The DOI matches its Crossref record; 133 of the 143 references it cites were found in a registry or an index; 5 of the 94 sources its claims cite were checked for support.
The geometry of G-structures
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CHERN 1. Introduction.Differential geometry studies differentiate manifolds and geometric objects or structures on them.It is now customary to distinguish it from differential topology by the presence of a structure in addition to the differentiable structure.What a differential geometric structure is or should be is a matter of taste.At the present state of the field a definition general enough to include all the significant structures will certainly contain many uninteresting ones.Among the attempts at a general definition is the notion of a geometric object initiated by Oswald Veblen [l30].Not all the geometrical structures are "equal".It would seem that the riemannian and complex structures, with their contacts with other fields of mathematics and with their richness in results, should occupy a central position in differential geometry.A unifying idea is the notion of a G-structure, which is the modern version of a local equivalence problem first emphasized and exploited in its various special cases by Elie Cartan [36], [41], [129].Generally we will restrict ourselves in this article to the discussion of problems which fall under the notion of a G-structure.Two general problems are of importance: I. Existence or nonexistence of certain structures on a manifold.EXAMPLE 1.A positive definite riemannian structure always exists.EXAMPLE 2. On a compact manifold M a nowhere zero differentiable vector field exists if and only if the Euler-Poincaré characteristic of M is zero.EXAMPLE 3. One may ask whether a nonzero vector field exists on M (supposed to be compact and orientable) which is parallel with respect to a riemannian metric.By Hodge's harmonic forms a necessary condition is that the first Betti number b l of M is ^ 1.One can further prove that the second Betti number b 2 ^bl -1 (cf.§5).These conditions are probably not sufficient.II. Local and global properties of a given structure.EXAMPLE 1.For a riemannian structure this means riemannian geometry.EXAMPLE 2. If we are only interested in the existence or nonexist-i 9 66] i 9 66]
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70 No basis given · 60 Sources not found · 31 Own interpretation · 11 Not checked yet · 10 No full text of the sources · 5 Couldn’t confirm · accuracy not yet measured
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“In the real analytic case the theorem of Cartan-Kuranishi [86] says that in a finite number of steps the system can be prolonged either to a system without solution or to a system in involution.”
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- Could not verifycites [86] 86. K. Kodaira and D. C. Spencer, Groups of complex line bundles over compact Kdhler varieties, Proc. Nat. Acad. Sci. U.S.A. 39 (1953), 868-872. 83 # sourceExperimental — accuracy not yet measured
The paper’s sections on complex line bundles and its stated theorem about Picard varieties address different subjects. Searches for the Cartan–Kuranishi theorem, real-analytic prolongation, and involutive systems found no passage addressing the cited claim; page numbers were not available in our copy. (LLM, conf=0.76)
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“A systematic theory of characteristic classes begins with the universal bundle theorem.It says that with a given compact manifold M as base space the fiber bundles with a structural group G (Lie group) are in one-one correspondence with the homotopy classes of mappings of M into a classifying space B G ; the latter depends only on G and on the dimension of M.”
§ X = Xi A * X + 7T2 A * 2 , > 9. Characteristic classes [12], [25], [64], [96].
What the sources say
- Could not verifycites [12] 12. M. F. Atiyah and I. M. Singer, The index of elliptic operators on compact mani- folds, Bull. Amer. Math. Soc. 69 (1963), 421-433. sourceExperimental — accuracy not yet measured
Theorem 2 (page unavailable) mentions the cohomology of a classifying space for G, but does not state the universal bundle theorem. Searches for the claimed one-to-one correspondence and its dependence on the dimension of M found no passage addressing either point. (LLM, conf=0.77)
- Not checked — no full text of the sourcecites [25] 25. A. Borel and J.-P. Serre, Groupes de Lie et puissances réduits de Steenrod, Amer. J. Math. 75 (1953), 409-448. source
- Not checked — no full text of the sourcecites [64] 64. N. J. Hicks, Notes on differential geometry, Math. Studies No. 3, Van Nos- trand, New York, 1965.
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“T. Banchoff gave examples of two polyhedral surfaces of genus one in euclidean three-space E z such that their corresponding faces are congruent, but they do not differ by a motion or a reflection of E z [14].”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
What the sources say
- Could not verifycites [14] 14. M. Berger, Sur quelques variétés riemanniennes suffisamment pincées, Bull. Soc. Math. France 88 (1960), 57-71. sourceExperimental — accuracy not yet measured
The Introduction states a theorem about pinched Riemannian manifolds, and the passages on closed geodesics concern those manifolds. Searches for Banchoff and for polyhedral surfaces with congruent faces found no passage addressing the attributed examples. (LLM, conf=0.76)
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“Then a study of the multiplicative properties of harmonic forms gives the theorem: r{M) = hi -h 2 .”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
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“Let M have a holonomic G-structure.Then J c VA = AiV.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 2. G = SO(p)XSO(n-p), l^p^n-l.
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“Again by considering the second variation of arc length Myers proved the following theorem [98]: A complete riemannian manifold with Ricci curvature = c ( = constant) > 0 is compact.”
§ THE GEOMETRY OF G-STRUCTURES 1 > •a
What the sources say
- Not matchedcites [98] 98. P. Liebermann, Problèmes d'équivalence relatifs à une structure presque com- plexe sur une variété à quatre dimensions, Acad. Roy. Belg. Bull. Cl. Sci. (5) 36 (1950), 742-755.
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“In the case of the frame bundle of a manifold this implies that the bundle space is topologically parallelisable, i.e., there are n 2 +n linear differential forms in the space, which are everywhere linearly independent.”
§ THE GEOMETRY OF G-STRUCTURES 1 > This formula gives the relation between du, Ov in UC\ V.
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“A complete riemannian manifold with sectional curvature S 0 has a universal covering space homeomorphic to the euclidean space.”
§ THE GEOMETRY OF G-STRUCTURES 1 > 2. Riemannian structure.
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“Much work has been done recently on complete riemannian manifolds with sectional curvature bounded below by a positive constant.Such a manifold is always compact.”
§ THE GEOMETRY OF G-STRUCTURES 1 > •a
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“By Hodge's harmonic forms a necessary condition is that the first Betti number b l of M is ^ 1.”
§ THE GEOMETRY OF G-STRUCTURES 1 > 1. Introduction.
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“From the last result one concludes that the first Betti number of the manifold is zero.”
§ THE GEOMETRY OF G-STRUCTURES 1 > •a
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“The symmetric riemannian manifolds of rank 1 mentioned above are the complex or quaternionic projective spaces or the Cayley plane.”
§ THE GEOMETRY OF G-STRUCTURES 1 > •a
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“For n = 2 the manifold is homeomorphic to the sphere.”
§ THE GEOMETRY OF G-STRUCTURES 1 > •a
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“Returning to the general case, it follows from ( 12) and a simple extension argument that a connection always exists in a principal fiber bundle.”
§ THE GEOMETRY OF G-STRUCTURES 1 > This formula gives the relation between du, Ov in UC\ V.
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“The problem of local invariants of a G-structure is essentially the following problem of equivalence of Elie Cartan [33]:”
§ THE GEOMETRY OF G-STRUCTURES 1 > This formula gives the relation between du, Ov in UC\ V.
What the sources say
- Not matchedcites [33] 33. > £ es groupes de transformations continus, infinis, simples, Ann. Sci. École Norm Sup. Paris 26 (1909), 93-161 ( = Oeuvres complètes, Partie II, pp. 857- 925).
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“One can further prove that the second Betti number b 2 ^bl -1 (cf.§5).”
§ THE GEOMETRY OF G-STRUCTURES 1 > 1. Introduction.
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“Besides the examples mentioned above the only known simply connected spaces having a riemannian metric of positive curvature are two examples given by Berger [lô]: These are homogeneous spaces of dimensions 7 and IS respectively and are not homeomorphic to the sphere.”
§ THE GEOMETRY OF G-STRUCTURES 1 > •a
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- Not matchedcites [lô]
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“For general compact G there may be different ways of attaching the connection which will be characterized by different properties of the torsion form.”
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“Let M be a compact real analytic manifold which has a real analytic leaved structure of codimension one.Then the fundamental group TTI(M) is not finite.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not matchedcites [59] 59. R. Gunning and H. Rossi, Analytic functions of several complex variables, Prentice-Hall, Englewood Cliffs, N. J., 1965.
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“(2) Let u be the cohomology class determined by Q.The homomorphism H r ~2(M, R)->H r (M, R) defined by a-^aSJu ( = cup product of a, u), aÇ:H r ~2(M, R), is, for r^m, an isomorphism.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
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“It can also be interpreted geometrically as the condition for a certain hexagonal configuration to exist.”
§ THE GEOMETRY OF G-STRUCTURES 1 > This formula gives the relation between du, Ov in UC\ V.
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“Examples (due to Atiyah, private communication) exist of four-dimensional compact manifolds with nonzero index but with two linearly independent vector fields (nonparallel with respect to any riemannian metric).”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not matchedcites Atiyah, private communication
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“Suppose that a compact orientable manifold of dimension 4k has 2k linearly independent vector fields which are parallel with respect to a riemannian metric.Then its index is zero.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
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“(1) Every odd-dimensional Betti number is even.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
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“For p== 1, a further analysis of the situation gives the inequality 6 2 ^61 -1, b 1 and b 2 being respectively the one and two-dimensional Betti numbers.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
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“The structure can be interpreted as a three-web of plane curves in the sense of Blaschke [24], the curves being defined respectively by the equations (30) 0 1 = 0, 0 2 = 0, 0 1 + 0 2 = 0.”
§ THE GEOMETRY OF G-STRUCTURES 1 > This formula gives the relation between du, Ov in UC\ V.
What the sources say
- Not checked — no full text of the sourcecites [24] 24. A. Borel, Sur la cohomologie des espaces fibres principaux de groupes de Lie compacts et des espaces homogènes, Ann. of Math. 57 (1953), 115-207. source
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“The fundamental theorem on harmonic forms says that the linear mapping p: H q -+H q (M, R) ( = ^-dimensional cohomology space of M with real coefficients) defined by sending a harmonic form to its cohomology class in the sense of de Rham's theorem is a one-one isomorphism.”
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“Newlander and Nirenberg proved that such an almost complex structure is subordinate to a complex structure [101], [103].”
What the sources say
- Not checked — no full text of the sourcecites [101] 101. J. Milnor, Lectures on characteristic classes, mimeographed notes, Univ. of Princeton, Princeton, N.J., 1958 (to appear).
- Not checked — no full text of the sourcecites [103] 103. S. Myers, Riemannian manifolds with positive mean curvature, Duke Math. J. 8(1941), 401-404. source
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“In the latter case there may be a pseudo-group of local homeomorphisms, other than the identity, which leaves the structures invariant.”
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“It follows that an almost complex manifold whose torsion form vanishes identically is a complex manifold.”
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“It follows from the theorem that if rj is harmonic, then PwV is also harmonic.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 2. G = SO(p)XSO(n-p), l^p^n-l.
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“Since M is compact, every harmonic form is closed.”
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“No connection can be attached to a complex manifold from its complex structure alone.”
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“In the particular case of the riemannian structure the Levi-Civita connection is completely determined by the vanishing of the torsion form.”
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“The condition A = 0 is a necessary and sufficient condition that the curves can be locally mapped into three families of parallel straight lines.”
§ THE GEOMETRY OF G-STRUCTURES 1 > This formula gives the relation between du, Ov in UC\ V.
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“(3) The homomorphism H r (M, R)->H 2 ™~r{M, R), r<>rn, defined by a-+aSJu m ~r, aÇiH r (M, R), is a one-one isomorphism.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
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“From this theorem Hodge^ index theorem on compact Kàhler manifolds can be derived by purely algebraic considerations.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
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“In this case we can derive that the pdimensional Betti number is ^ 1.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
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“It can be proved that such a connection exists whenever G is compact [43].”
What the sources say
- Not matchedcites [43] 43. f On curvature and characteristic classes of a Riemann manifold, Abh. Math. Sem. Univ. Hamburg 20 (1955), 117-126.
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“By Frobenius' theorem this means that the pfaffian system”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not matchedcites Frobenius
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“Hence if A q (T*) is a direct sum of the invariant subspaces Wi, • • • , Wk, the space H q of harmonic forms of degree q is a direct sum of the spaces of harmonic forms of types Wi, • • • , Wk respectively.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 2. G = SO(p)XSO(n-p), l^p^n-l.
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In the paper
“The difficulty in proving this is the local theorem of Korn and Lichtenstein to the effect that a two-dimensional (positive definite) C°°-riemannian manifold is locally conformai to the euclidean plane.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not matchedcites Korn and Lichtenstein
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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In the paper
“On the other hand, 0(2m)/U(m)XT 2m has a Kâhlerian complex structure derived from those of the two factors.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“By a theorem of Blanchard [23] the compact complex manifold so obtained (the fiber space!) which, as a differentiable manifold, is the product 0(2m)/U(m) XT 2m (T 2m is a real torus of real dimension 2m), has no Kâhlerian structure.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not matchedcites [23] 23. W. Blaschke and G. Bol, Geometrie der Gewebe, Springer, Berlin, 1938.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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In the paper
“By applying this isomorphism successively, we get”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 5. DE RHAM'S THEOREM.
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“Then condition (35) is equivalent to the condition that the brackets [Xi, Xj] are linear combinations of Xi, • • • , X p .”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“As complex manifolds the S n are distinct from each other (a theorem of Hirzebruch [65]).”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not checked — no full text of the sourcecites [65] 65. F. Hirzebruch, Neue topologische Methoden in der algebraischen Geometrie, Ergebnisse der Math und ihrer Grenzgebiete, 1962. (55. 1 fiber eine Klasse von einfach-zusammenhiingenden komplexen Mannig- faltigkeiten, Math Ann. 124 (1957), 77-86. source
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In the paper
“K is simply connected (Spanier [127]).”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not matchedcites Spanier [127]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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In the paper
“A remarkable recent result of van de Ven, not yet published, gives an example of a compact four-dimensional almost complex manifold which has no complex structure.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not matchedcites van de Ven
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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In the paper
“We get in this way a manifold which has both Kàhlerian and non-Kâhlerian complex structures.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
plainSingle case- Asserted: passes — The authors put it forward as true.
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In the paper
“The essential point in the proof of this exactness is the so-called Poincaré lemma, which says that any closed differential form is locally an exterior derivative.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 5. DE RHAM'S THEOREM.
What the sources say
- Not matchedcites Poincaré
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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In the paper
“Then the sequence 0 -» C p -> A p -> C p+l -» 0, where i is the injection and d is the exterior differentiation, is exact.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 5. DE RHAM'S THEOREM.
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In the paper
“Grothendieck's theorem is then the formula: (64) f*(ch(W)t(M)) = chiftWXN).”
§ X = Xi A * X + 7T2 A * 2 , > \ il-exp(-Yi) /
What the sources say
- Not matchedcites Grothendieck
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“For the case of the tangent bundle of a manifold the reduction to a G'-bundle is equivalent to the existence of an almost complex structure.”
§ X = Xi A * X + 7T2 A * 2 , > Let x 0 be a nonzero vector through the origin of E 9+N (R) and perpendicular to E q+N ~1(R).
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“From the definition it follows that the characteristic classes have the naturality property, i.e., that they are contra variant functors under bundle maps.”
§ X = Xi A * X + 7T2 A * 2 , > 9. Characteristic classes [12], [25], [64], [96].
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“If M is compact, we get the exact sequence”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 5. DE RHAM'S THEOREM.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“It follows from the exact sequence of cohomology groups that this Cousin problem always has a solution if H l (M, 12) =0.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 5. DE RHAM'S THEOREM.
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“If 12* denotes the sheaf of germs of nonzero holomorphic functions, the group of complex line bundles over a complex manifold Mis H l (M, 12*).”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 5. DE RHAM'S THEOREM.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Thus the characteristic classes are the first invariants which describe the deviation of a fiber bundle from a product bundle.”
§ X = Xi A * X + 7T2 A * 2 , > 9. Characteristic classes [12], [25], [64], [96].
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“H r (M, B*'°) 9Ë H°(M, B*> r )/d”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 5. DE RHAM'S THEOREM.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“The linear equivalence class of div(co), to be denoted by K, is independent of œ and depends only on M, because the ratio of two meromorphic differential forms is a meromorphic function.”
§ X = Xi A * X + 7T2 A * 2 , > Divisibility.
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Then the sequence of sheaves i à”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 5. DE RHAM'S THEOREM.
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“In fact, Chow's theorem implies that if V is compact, the image set ƒ( V) of a holomorphic mapping/: V-*P m ( = projective space of dimension m) is an algebraic variety.”
§ X = Xi A * X + 7T2 A * 2 , > 11. Holomorphic mappings of complex analytic manifolds [l], [45], [137].
What the sources say
- Not matchedcites [58] 58. D. Gromoll, Differenzierbare Strukturen und Metriken positiver Krümmung auf Sphàren, Dissertation, Univ. of Bonn, Bonn, 1964.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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In the paper
“An important divisibility property is given by the following theorem of Bott [28 ] : The characteristic number c n of any complex vector bundle over the sphere S 2n is divisible by (n -1) !.”
§ X = Xi A * X + 7T2 A * 2 , > Divisibility.
What the sources say
- Not matchedcites [28] 28. j The stable homotopy of the classical groups, Ann. of Math. 70 (1959), 313-337.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Hirzebruch's formula says that, for nonsingular algebraic varieties, we have (58) X (M, W) = T(M, W).”
§ X = Xi A * X + 7T2 A * 2 , > \ il-exp(-Yi) /
What the sources say
- Not matchedcites Hirzebruch
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“(1) The additive Cousin problem always has a solution on a Stein manifold.”
§ X = Xi A * X + 7T2 A * 2 , > Stein manifolds.
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“Similarly, we have, for an 0(g)-bundle or an SO (g)-bundle, the characteristic classes W'^H^M, Z 2 ), Orgigg, and p k <EH* k (M, Z), 0£*£[g/4], called respectively the Stief el-Whitney classes and the Pontrjagin classes.”
§ X = Xi A * X + 7T2 A * 2 , > 9. Characteristic classes [12], [25], [64], [96].
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“The generalization by Atiyah-Hirzebruch is the following theorem.Let M and N be compact oriented differentiable manifolds with W*{M) = W*(N) = 0 and dim Af-dim N=0, mod 2. Letf: M->N be a continuous mapping.Then f*(R(M)) QR(N).”
§ X = Xi A * X + 7T2 A * 2 , > MR(M)) C R(N),
What the sources say
- Not matchedcites Atiyah-Hirzebruch
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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In the paper
“The cohomology groups of these Grassmann manifolds have been determined by Ehresmann.”
§ X = Xi A * X + 7T2 A * 2 , > 9. Characteristic classes [12], [25], [64], [96].
What the sources say
- Not matchedcites Ehresmann
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“A complex manifold M is a Stein manifold if and only if the following two properties hold : (A) For any coherent sheaf S over M the module of global sections of S over M generates at every point xÇzM the module of local sections. (B) For any coherent sheaf S over M, H q (M, S) =0, q^ 1.”
§ X = Xi A * X + 7T2 A * 2 , > Stein manifolds.
What the sources say
- Not matchedcites Oka, Cartan, and Serre
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“By introducing an hermitian scalar product in the vector bundle W, Kodaira extended Hodge's harmonic forms and proved that H q (M, A* (WO) is isomorphic to the space H p > q of harmonic forms of type (p, q) and with values in W [79].”
§ X = Xi A * X + 7T2 A * 2 , > Compact complex manifolds.
What the sources say
- Not matchedcites [79] 79. III, J. Math. Mech. 14 (1965), 679-706;
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“This is imossible for w^4, which gives a new proof of the Borel-Serre theorem that S 2n has no almost complex structure for n ^ 4.”
§ X = Xi A * X + 7T2 A * 2 , > Divisibility.
boosteddirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Frow this Hirzebruch's formula follows by taking N to be a point.”
§ X = Xi A * X + 7T2 A * 2 , > \ il-exp(-Yi) /
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“A theorem of Peterson says that if M has in dimension 2k only torsion coefficients which are relatively prime to (k -1) !, then a complex vector bundle of sufficiently large dimension over M is a product bundle if all its characteristic classes are zero”
§ X = Xi A * X + 7T2 A * 2 , > Divisibility.
What the sources say
- Not matchedcites Peterson
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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In the paper
“In particular, there are elements of H 2i (G(q y N, C), Z), O^igg, which can, for instance, be completely described by the Schubert varieties and which generate the cohomology ring H*(G(q, N, C), Z), such that their images under/* are the characteristic classes Ci€:H 2i (M, Z) of the Z7(<z)-bundle.”
§ X = Xi A * X + 7T2 A * 2 , > 9. Characteristic classes [12], [25], [64], [96].
What the sources say
- Not matchedcites Ehresmann
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“Another important consequence of the theorem is that the Todd genus of a nonsingular algebraic variety is a birational invariant.”
§ X = Xi A * X + 7T2 A * 2 , > \ il-exp(-Yi) /
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“The structure function is zero for a locally flat structure.”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Without analyticity the same result has only been proved for n = 2, and in fact only under the additional assumption that the gaussian curvature of V keeps a constant sign in a neighborhood [61].”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
What the sources say
- Not matchedcites [61] 61. Variétés feuilletées, Ann. Scuola Norm. Sup. Pisa. 16 (1962), 367-397.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plainn = 2- Asserted: passes — The authors put it forward as true.
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- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Then Plücker's formulas are (70) -w, -vp-x + 2v p -v p+1 = x(V), Ogpgm-1.”
§ X = Xi A * X + 7T2 A * 2 , > 11. Holomorphic mappings of complex analytic manifolds [l], [45], [137].
plaincorrelational- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“When dim F>1, value distribution in the strict sense fails completely.”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
plaindirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“For an almost complex structure the vanishing of the structure function is equivalent to the fulfillment of the integrability conditions.”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
plaincorrelational- Asserted: passes — The authors put it forward as true.
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None of the sources cited for this claim have been checked yet.
In the paper
“This theorem has a generalization to minimal surfaces in E m”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
What the sources say
- Not checked — no full text of the sourcecites [47] 47. S. S. Chern, Complex analytic mappings of Riemann surfaces. I, Amer. J. Math. 82 (1960), 323-337. source
- Not matchedcites [107] 107. I, Jber. Deutscher. Math.-Verein 63 (1960-1961), 117-136. 132. H. C. Wang, Two-point homogeneous spaces, Ann. of Math. 55 (1952), 177- 191. 133 #
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plain- Asserted: passes — The authors put it forward as true.
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In the paper
“A surface in E m is minimal, if and only if the Gauss mapping, which sends a point p of the surface to the plane through 0 parallel to the tangent plane at p, is anti-holomorphic.”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“This formula, to be called the first main theorem, is due to H. Levine [92].”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
What the sources say
- Not checked — no full text of the sourcecites [92] 92. M. Kuranishi, On E. Cartan's prolongation theorem of exterior differential sys- tems, Amer. J. Math. 79 (1957), 1-47. §7 # f Q n the locally complete families of complex analytic structures, Ann. of Math. 75 (1962), 536-577. gg # source
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Then the set of linear spaces of dimension m-n which do not meet ƒ( V) is of measure zero in the Grassmann manifold Gr(w, m-n) of linear sub spaces of dimension m-n in P m , m*£n.”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
What the sources say
- Not matchedcites [46] 46. II, Michigan Math. J 5 (1958), 5-12.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- Asserted: passes — The authors put it forward as true.
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- Not matched
- Probably not a substantial claim. 88
- The quote only partly states this claim. 2
- It may state two separate claims as one. 28
The cited sources that could be checked couldn’t confirm or dispute this claim.
In the paper
“The Euler-Poincaré characteristic of a compact orientable riemannian manifold M of even dimension n = 2m is given by the Gauss-Bonnet formula [4], [39],”
§ THE GEOMETRY OF G-STRUCTURES 1 > 2. Riemannian structure.
What the sources say
- Could not verifycites [4] 4. W. Ambrose and I. M. Singer, A theorem on holonomy, Trans. Amer. Math. Soc. 75 (1953), 428-443. sourceExperimental — accuracy not yet measured
The Notation section discusses Riemannian curvature and the holonomy group, but does not relate curvature to the Euler–Poincaré characteristic. Searches of our full-text copy for Gauss–Bonnet and Euler characteristic found no passage stating the claimed formula. (LLM, conf=0.76)
- Not matchedcites [39] 39. Thone, Liège, 1953.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Could not verify
The cited sources that could be checked couldn’t confirm or dispute this claim.
In the paper
“In high dimensions Frölicher and Nijenhuis proved that the complex structure on a compact complex manifold M is rigid, if H l (M, ®)=0, where H X (M, ©) is the one-dimensional cohomology group of M over the sheaf of germs of holomorphic vector fields (cf.§8) [53].”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Could not verifycites [53] 53. T. Frankel, Manifolds with positive curvature, Pacific J. Math. 11 (1961), 165- 174. sourceExperimental — accuracy not yet measured
The introduction and §4 discuss positive curvature and Kähler surfaces; §4 includes a cohomology vanishing result, but not the claimed rigidity criterion. Searches for Frölicher and Nijenhuis and for rigidity under H¹(M,Θ)=0 found no passage addressing the claim. Page numbers were not available in our copy. (LLM, conf=0.78)
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Could not verify
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“One sees that X is the gaussian curvature of the riemannian structure.”
§ THE GEOMETRY OF G-STRUCTURES 1 > This formula gives the relation between du, Ov in UC\ V.
plaincorrelational- Asserted: fails — The authors may not put it forward as true.
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Probably not a substantial claim. The authors may not put it forward as true. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. The quote only partly states this claim.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Moreover, if Q is the multiplication by an invariant exterior differential form, we have also ÇA=A<2.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 2. G = SO(p)XSO(n-p), l^p^n-l.
plaindirect- Asserted: passes — The authors put it forward as true.
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The quote only partly states this claim.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In fact, the complex projective space with the usual elliptic hermitian metric has sectional curvatures R satisfying the inequalities A/4t^R ^Ay A being a positive constant.”
§ THE GEOMETRY OF G-STRUCTURES 1 > •a
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“According to a theorem of H. C. Wang [132], these, together with the spherical and elliptic spaces, are the only compact and connected two-point homogeneous spaces.”
§ THE GEOMETRY OF G-STRUCTURES 1 > •a
What the sources say
- Not matchedcites [132]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Parallelism is at the basis of the notion of curvature, for geometrically curvature measures the dependence of parallelism on the curve.”
§ THE GEOMETRY OF G-STRUCTURES 1 > •a
plaindirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“For high dimensions constancy of the sectional curvature will lead to the space forms and it is known that not every manifold has a riemannian metric of constant curvature.”
§ THE GEOMETRY OF G-STRUCTURES 1 > •a
boosteddirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Rauch proved the following theorem : A complete simply connected a-pinched riemannian manifold with a = 0.75 is homeomorphic to the sphere,”
§ THE GEOMETRY OF G-STRUCTURES 1 > •a
plaindirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“When the riemannian manifold is a surface in ordinary euclidean space, the parallelism along a curve is obtained by taking the tangent developable surface and rolling it on a plane.”
§ THE GEOMETRY OF G-STRUCTURES 1 > •a
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“A result in this direction is the following theorem of Yamabe [l40]: A compact riemannian manifold of dimension ^3 is conformally equivalent to one of constant scalar curvature.”
§ THE GEOMETRY OF G-STRUCTURES 1 > •a
What the sources say
- Not matchedcites [l40]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“As an Einstein metric on a three-dimensional manifold is necessarily of constant curvature, the problem has a bearing on the Poincaré hypothesis that a compact simply connected three-dimensional manifold is homeomorphic to the sphere.”
§ THE GEOMETRY OF G-STRUCTURES 1 > •a
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“The statement is true for w = 4 [42] and for the case that M has constant sectional curvature.”
§ THE GEOMETRY OF G-STRUCTURES 1 > ij '
What the sources say
- Not checked yetcites [42] 42. Pseudo-groupes continus infinis. Geometrie différentielle, Colloquium, Strasbourg, pp. 119-136, 1953, Centre National de la Recherche Scientifique, Paris, 1953.
plainn = 4- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“An 0(n)-structure is nothing else but a riemannian structure.”
§ THE GEOMETRY OF G-STRUCTURES 1 > This formula gives the relation between du, Ov in UC\ V.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“It follows that its universal covering manifold is also compact and hence that its fundamental group is finite.”
§ THE GEOMETRY OF G-STRUCTURES 1 > •a
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“The homeomorphism theorem of Klingenberg and Berger was improved by D. Gromoll [57] and E. Calabi to a diffeomorphism theorem : There exists a number ô n depending onn (0<8 n < 1, lim^,» ô n = 1) such that if a simply connected riemannian manifold M of dimension n is 8 n -pinched, then M is diffeomorphic to the n-sphere with its usual differentiable structure.”
§ THE GEOMETRY OF G-STRUCTURES 1 > •a
What the sources say
- Not checked — no full text of the sourcecites [57] 57. M. Goto, On algebraic homogeneous spaces, Amer. J. Math. 76 (1954), 811-818. source
plaindirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The classical example of a connection is the parallelism of Levi-Civita in riemannian geometry.”
§ THE GEOMETRY OF G-STRUCTURES 1 > •a
plain- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The curvature forms define the local properties of a connection and the holonomy groups iJ& its global property.”
§ THE GEOMETRY OF G-STRUCTURES 1 > This formula gives the relation between du, Ov in UC\ V.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“Ambrose and Singer showed how the Lie algebra of H& can be determined from the curvature forms [5].”
§ THE GEOMETRY OF G-STRUCTURES 1 > This formula gives the relation between du, Ov in UC\ V.
What the sources say
- Not checked — no full text of the sourcecites [5] 5. A. Andreotti, On the complex structures of a class of simply-connected manifolds. Algebraic geometry and topology (Lefschetz Jubilee Volume), Princeton Univ. Press, Princeton, N. J., 1957, pp. 53-77. source
plaindirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“For n = 2rn consider GL(m, C) as a subgroup of GL(w, R).Then a GL(m, C)-structure is what is called an almost complex structure.”
§ THE GEOMETRY OF G-STRUCTURES 1 > This formula gives the relation between du, Ov in UC\ V.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Synge's lemma says the following: Let M be a simply connected compact even-dimensional riemannian manifold whose sectional curvature is >0.Let g be a closed geodesic of minimum length among the closed geodesies of M. Then there is a family of closed curves converging toward g, which are all shorter than g.”
§ THE GEOMETRY OF G-STRUCTURES 1 > •a
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“For a compact two-dimensional manifold this will be one of constant gaussian curvature; the curvature is positive for the sphere, zero for the torus, and negative for a surface of genus >1.”
§ THE GEOMETRY OF G-STRUCTURES 1 > •a
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“Andreotti and Frankel proved that if a compact Kàhler manifold of complex dimension 2 has positive sectional curvature, it is complex analytically equivalent to the complex projective plane [52].”
§ THE GEOMETRY OF G-STRUCTURES 1 > •a
What the sources say
- Not checked yetcites [52] 52. Thone, Liège, 1951. 51. 1? $ ur i a théorie des variétés feuilletées, Univ. Roma 1st. Naz. Alta Rend Mat. e Appl. (5) 10 (1951), 64-82.
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“A compact riemannian manifold with sectional curvature <0 has a fundamental group which cannot be abelian and of which every abelian subgroup is cyclic.”
§ THE GEOMETRY OF G-STRUCTURES 1 > 2. Riemannian structure.
What the sources say
- Not matchedcites [112]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
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- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“It applies to the high-dimensional Betti numbers of riemannian manifolds and, more importantly, to the derivation of criteria (in terms of curvature properties) for the vanishing of more general cohomology groups (cf.§8).”
§ THE GEOMETRY OF G-STRUCTURES 1 > •a
plaindirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“EXAMPLE 7. G is the group of all matrices of order n ( è 3) of the form X^4, X>0, where A is orthogonal.This is the conformai structure, to which a normal conformai connection, uniquely determined, can be attached.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“The problem of equivalence was solved by Libermann for m = 2 [93].”
What the sources say
- Not matchedcites [93] 93. ew proof for the existence of locally complete families of complex struc- tures, pp. 142-154, Proc. Conf. Comp. Anal., Minneapolis, 1964, Springer, Berlin, 1965.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- Asserted: passes — The authors put it forward as true.
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- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The structure is given by (0 1 ) 2 + (0 2 ) 2 , determined up to a positive factor.It is therefore the two-dimensional conformai structure.”
§ THE GEOMETRY OF G-STRUCTURES 1 > This formula gives the relation between du, Ov in UC\ V.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Then a G-structure on M defines an orientation of M and a riemannian structure, to be called the associated riemannian structure.”
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“In particular, Kaplan proved that to a leaved structure in the plane there exists a continuous realvalued function in the plane, which has neither a maximum nor a minimum and which is constant on the leaves.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not matchedcites [69] 69. W. Kaplan, Regular curve-families filling the plane, I, Duke Math. J. 7 (1940), 154-185;
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“This means that the manifold has a field of ^-dimensional plane elements parallel with respect to the Levi-Civita connection.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 2. G = SO(p)XSO(n-p), l^p^n-l.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“This is essentially the Kàhler structure on a complex manifold in its real formulation.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“EXAMPLE 6. G -0(n).This is the riemannian structure where the Levi-Civita connection can be attached.”
plaincorrelationaln = 2- Asserted: passes — The authors put it forward as true.
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In the paper
“They correspond respectively to the cases that the two G-structures are locally inequivalent or equivalent.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“A notable example of a leaved structure of codimension one on the three-sphere S 3 was given by G. Reeb [116].”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not matchedcites [116]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- Asserted: passes — The authors put it forward as true.
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- Not matched
In the paper
“The fundamental theorem of Denjoy says that (under sufficient smoothness hypotheses) if there is no closed leaf, then every leaf is ergodic.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not matchedcites Denjoy
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
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- Not matched
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“Such a G-structure is called holonomic, if it satisfies one of the following three conditions, which are equivalent [43], [84], [135]:”
What the sources say
- Not matchedcites [43] 43. f On curvature and characteristic classes of a Riemann manifold, Abh. Math. Sem. Univ. Hamburg 20 (1955), 117-126.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [84] 84. f On a differential geometric method in the theory of analytic stacks, Proc. Nat. Acad. Sci. U.S.A. 39 (1953), 1268-1273.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [135]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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- Not matched
In the paper
“A consequence of the Poincaré-Bendixon theory says that the leaves go to infinity in both directions.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not matchedcites Poincaré-Bendixon
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
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- Not matched
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“The Den joy-Schwartz theorem says that if a compact connected two-dimensional surface is under the action of the real line, then a minimal set is: (1) either a single point; (2) or a closed curve; (3) or the whole surface, the last case happening only for the torus.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not matchedcites Denjoy
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites Schwartz
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“No connection can be introduced, which will depend on this structure alone; for otherwise it would mean that the local homeomorphisms leaving the conformai structure invariant would depend on a finite number of constants.”
§ THE GEOMETRY OF G-STRUCTURES 1 > This formula gives the relation between du, Ov in UC\ V.
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Peixoto proved that the structurally stable vector fields on a compact two-dimensional surface are open and dense in the set of all vector fields on the surface [109 ].”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not checked yetcites [109] 109. Selecta Her- mann Weyl, pp. 148-178, Birkhâuser Verlag, Basel, 1956. 139. W. T. Wu, Sur les classes caractéristiques des structures fibrées sphériques, Actualités Sci. Indust., Hermann, Paris, 1952. 140. H. Yamabe, On deformation of riemannian structures on compact manifolds, Osaka Math. J. 12 (1960), 21-37. 141. K. Yano and S. Bochner, Curvature and Betti numbers, Annals of Mathe- matics Studies No. 32, Princeton Univ. Press, Princeton, N. J., 1953.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Thus S 2 XS 2 is a simply connected manifold which admits an infinite number of complex structures.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Blanchard proved that if a compact Kâhler manifold is blown up along a nonsingular submanifold, the resulting manifold is again Kâhlerian [23].”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not matchedcites [23] 23. W. Blaschke and G. Bol, Geometrie der Gewebe, Springer, Berlin, 1938.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“Andreotti proved that there are no other algebraic structures [6].”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not matchedcites [6] 6. A. Andronov and L. Pontrjagin, Systèmes grossiers, C.R. Acad. Sci. Paris 14 (1937), 247-250.
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In the paper
“Riemann's mapping theorem says that S 2 has a uniquely determined complex structure.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not matchedcites Riemann
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In the paper
“Andreotti also proved that there is only one Kàhler structure on P 2 (the standard one); Hirzebruch and Kodaira proved that there is only one Kâhler structure on P m for odd m [67].”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not matchedcites [67] 67. F. Hirzebruch and K. Kodaira, On the complex projective spaces, J. Math. Pures Appl. 36 (1957), 201-216.
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In the paper
“This was generalized by Lima and H. Rosenberg to the result that the rank of a compact simply connected manifold of dimension n is at most n -2.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not matchedcites Lima and H. Rosenberg
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In the paper
“If a manifold has a leaved structure of dimension p, it must have a G-structure, with G = GL(p, R).”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
plaindirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Hence H°(M, S) is the group of all global sections of S.”
§ X = Xi A * X + 7T2 A * 2 , > 0-~>R~>S-+T->0
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“The crowning achievement is the theorem of M. Kuranishi, which says that, for any compact complex manifold, there exists a universal family of deformations”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not matchedcites Kuranishi
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In the paper
“For m> 1 a Hopf manifold cannot be given a Kâhler structure and is therefore not algebraic, because the latter must have its second Betti number ^1.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“Compact submanifolds of P m are, by a theorem of Chow, the same as the nonsingular algebraic varieties.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not matchedcites Chow
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In the paper
“Goto, Borel, and Weil proved that it is an algebraic variety and Goto proved that it is rational [56], [124].”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not checked — no full text of the sourcecites [56] 56. S. I. Goldberg, Curvature and homology, Academic Press, New York, 1962.
- Not matchedcites [124]
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In the paper
“If G is a connected compact Lie group and T a maximal toroid of G, Borel observed that G/T has a complex structure [25].”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not checked — no full text of the sourcecites [25] 25. A. Borel and J.-P. Serre, Groupes de Lie et puissances réduits de Steenrod, Amer. J. Math. 75 (1953), 409-448. source
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In the paper
“In fact, Smale gave examples of manifolds in which the structurally stable vector fields are not dense in the set of all vector fields.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not matchedcites Smale
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In the paper
“Lima proved that S z has rank one [95].”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not matchedcites [95] 95. 1 Introduction to differentiable manifolds, Interscience, New York, 1962.
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In the paper
“Let”
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In the paper
“Among its consequences is the theorem that the closure of the set of all nonwandering points is the closure of the set of all closed trajectories.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not matchedcites [113]
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In the paper
“If 5 is the sheaf of germs of mappings into a topological group Y (F not necessarily abelian), H l (M, S) is the set of all equivalence classes (in the sense of bundles) of bundles with the group V.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 4. Meaning of H l (M, S).
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“Wang determined all the compact homogeneous complex manifolds with a finite fundamental group, among which are the simply connected even-dimensional compact Lie groups [133].”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
What the sources say
- Not matchedcites [133]
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Topologically it is homeomorphic to S 1 XS 2m ~l.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“A homomorphism of sheaves S-+S' induces a homomorphism H q (M, S)^>H q (M, S') of the corresponding cohomology groups.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“An orientable manifold of two (real) dimensions always has a complex structure.”
§ X = Xi A * X + 7T2 A * 2 , > EXAMPLE 1 (CONTINUED).
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“The Atiyah-Singer theorem says that it{D) =i a (D).”
§ X = Xi A * X + 7T2 A * 2 , > The Atiyah-Singer theorem says that it{D) =i a (D).
What the sources say
- Not matchedcites Atiyah-Singer
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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In the paper
“(2) The second Cousin problem, the problem whether a given divisor is the divisor of a meromorphic function, has a solution on a Stein manifold if H*(M, Z)=0.”
§ X = Xi A * X + 7T2 A * 2 , > Stein manifolds.
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“(3) Every meromorphic function on a Stein manifold is the quotient of two holomorphic functions.”
§ X = Xi A * X + 7T2 A * 2 , > Stein manifolds.
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Actually Serre's duality theorem is valid for noncompact complex manifolds; we state it for compact manifolds for simplicity.”
§ X = Xi A * X + 7T2 A * 2 , > Compact complex manifolds.
What the sources say
- Not matchedcites [122]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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In the paper
“the classical Riemann-Roch theorem is given by the formula ( 51) 1(D) = d(D) -g + i + 1, where g is the genus of M and i à 0 is the index of specialty of D.”
§ X = Xi A * X + 7T2 A * 2 , > Divisibility.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“Serre's duality theorem [122] says that the vector spaces H«(M, A r (W)) and H m ~q(M, A m ~r(W*)) are in duality and hence have the same dimension.”
§ X = Xi A * X + 7T2 A * 2 , > Compact complex manifolds.
What the sources say
- Not matchedcites [122]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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In the paper
“A theorem of Weil states that the cohomology class determined by F($) is independent of the choice of the connection and depends only on the bundle [40 ].”
§ X = Xi A * X + 7T2 A * 2 , > Let x 0 be a nonzero vector through the origin of E 9+N (R) and perpendicular to E q+N ~1(R).
What the sources say
- Not checked — no full text of the sourcecites [40] 40. S. S. Chern, On the curvatura intégra in a riemannian manifold, Ann. of Math. 46 (1945), 674-684. source
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“By studying this mapping h one derives easily that a necessary condition for the reduction of the G-bundle to a G'-bundle is that the Euler class should be zero.(In this case it can be proved that the condition is also sufficient.)”
§ X = Xi A * X + 7T2 A * 2 , > Let x 0 be a nonzero vector through the origin of E 9+N (R) and perpendicular to E q+N ~1(R).
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“When M is compact, Cartan and Serre proved that H q (M, S) is of finite dimension for any analytic coherent sheaf 5.”
§ X = Xi A * X + 7T2 A * 2 , > Compact complex manifolds.
What the sources say
- Not matchedcites Cartan and Serre
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In the paper
“Then the Riemann-Roch theorem can be written (54) dim H°(M, Q(D)) -dim E l (M, ti(D)) = d -g + 1.”
§ X = Xi A * X + 7T2 A * 2 , > Divisibility.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“Conversely, it was proved by Kodaira and Spencer that every line bundle over a nonsingular algebraic variety is defined by a divisor.”
§ X = Xi A * X + 7T2 A * 2 , > Divisibility.
What the sources say
- Not matchedcites Kodaira and Spencer
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
boostedcorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“He, and later Akizuki and Nakano, gave sufficient conditions of a differential-geometric nature for the vanishing of the cohomology groups H q (M, A P (W)) (for the particular case of a line bundle W) [2], [80 ].”
§ X = Xi A * X + 7T2 A * 2 , > Compact complex manifolds.
What the sources say
- Not checked — no full text of the sourcecites [2] 2. A. D. Alexandrow, Die innere Geometrie der konvexen Flâchen, Akademie Verlag, Berlin, 1955. source
- Not matchedcites [80] 80. IV, J. Math. Mech. 15 (1966), 163-175;
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In the paper
“Kodaira's theorem says that if F is a Kâhler manifold whose fundamental two-form has rational periods, then there exists a holomorphic mapping/: V->P m which is one-to-one.”
§ X = Xi A * X + 7T2 A * 2 , > 11. Holomorphic mappings of complex analytic manifolds [l], [45], [137].
What the sources say
- Not matchedcites [81] 81. V, (to appear).
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In the paper
“A classical relation of this nature is the Riemann-Hurwitz formula: Let V, M be Riemann surfaces with Euler characteristics x(^0> x(^f) respectively.Let ƒ : V-+M be a holomorphic mapping of degree d.Then we have (69) x(V) + w = d x (M), where w is the index of ramification, i.e., the sum of the orders of the points of ramification.”
§ X = Xi A * X + 7T2 A * 2 , > 11. Holomorphic mappings of complex analytic manifolds [l], [45], [137].
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“Applying the results of Grothendieck and Thorn, I. R. Porteous [ill] derived relations between the characteristic classes for the following cases: (a) dilatations; (b) ramified coverings with singularities of a relatively simple type.”
§ X = Xi A * X + 7T2 A * 2 , > 11. Holomorphic mappings of complex analytic manifolds [l], [45], [137].
What the sources say
- Not matchedcites [111]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“A complete minimal submanifold in E m is never compact.”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“For a C^-riemannian manifold V of dimension n Nash's theorem, improved by N. H. Kuiper, says that V can be CMsometrically imbedded in an euclidean space of dimension 2n+1 [85], [99].”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
What the sources say
- Not matchedcites [85] 85. On Kdhler varieties of restricted type, Ann. of Math. 60 (1954), 28-48.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [99] 99. A. Lichnerowicz, Théorie globale des connexions et des groupes d'holonomie, Edizioni Cremonese, Rome, 1957.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
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- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Cartan's determination of the simple infinite pseudo-groups (over the complex field) has not been completely verified”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“For example, Nash proved that a compact two-dimensional riemannian manifold can be isometrically imbedded in an euclidean space of dimension 17.”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“If an isometry is present, the basic local invariant is the second fundamental form, which is a quadratic differential form, with value in the normal bundle.”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
None of the sources cited for this claim have been checked yet.
In the paper
“A. D. Alexandrow proved that two tightly imbedded analytic compact surfaces of genus one in euclidean three-space E z differ by an isometry of E z , if they are isometric.”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“If V and M are real analytic riemannian manifolds of dimensions n and tn = n(n + l)/2 respectively, then the theorem of Janet-Cartan says that there is locally an isometric mapping of V into M [35].”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
What the sources say
- Not matchedcites [35] 35. Oeuvres complètes, Partie III, pp. 1091-1097).
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
None of the sources cited for this claim could be matched to a paper in Crossref, so there is nothing to check it against.
In the paper
“Nash's imbedding theorem is still true, if more smoothness is imposed, but the dimension of the receiving euclidean space will have to be higher and the proof is more difficult [89], [97], [lOO].”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
What the sources say
- Not matchedcites [89] 89. V. Yoh Kraines, Topology of quaternionic manifolds, Bull. Amer. Math. Soc. 71 (1965), 526-527.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [97] 97. H. Le vine, A theorem on holomorphic mappings into complex projective space, Ann. of Math. 71 (1960), 529-535.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not matchedcites [lOO]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The study of the zeroes of a section or of points at which several sections are linearly dependent is closely related to the value distribution problem and is in many cases equivalent to it.”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The second fundamental form is related to the curvature forms of both riemannian manifolds by the (generalized) Gauss equations, but until now the exact relationship (a purely algebraic problem) has not been sufficiently clarified.”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
In the paper
“Much sharper results generalizing Picard's theorem were obtained by R. Nevanlinna in the form of his defect relations.”
§ X = Xi A * X + 7T2 A * 2 , > II. Equidistribution.
What the sources say
- Not matchedcites Nevanlinna
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
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- Not matched
In the paper
“From it an equidistribution theorem was derived, but much remains to be done.”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
What the sources say
- Not checked — no full text of the sourcecites [30] 30. E. Calabi, On compact riemannian manifolds with constant curvature. I, pp. 155-180, Differential geometry, Proc. Sympos. Pure Math., Vol. 3, Amer. Math. Soc, Providence, R. I., 1961. source
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“A classical theorem of Cohn-Vossen, complemented by a remark of K. Voss, says that a closed convex surface (i.e., Gaussian curvature ^0) in an euclidean threespace E z is completely determined by its metric, modulo the isometriesof E z [131 ].”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
What the sources say
- Not matchedcites [131]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
boosteddirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“For a given compact manifold the minimum of the total curvatures for all possible immersions is a differential topological invariant and is an integer.”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In fact, the classical example of Fatou and Bieberbach is a holomorphic mapping/: JE 2 ->E 2 , such that the Jacobian determinant is identically equal to one, while E^-fiE^) contains an open subset.”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
plain- Asserted: passes — The authors put it forward as true.
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- Load-bearing: fails — The paper’s conclusions may not depend on it.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“To remove the assumption of analyticity Nirenberg has to put in some additional hypotheses [105], so that the question whether the analyticity assumption in Alexandrow's theorem could be removed remains unsettled.”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
What the sources say
- Not checked — no full text of the sourcecites [105] 105. H. E. Rauch, A contribution to differential geometry in the large, Ann. of Math. 54(1951), 38-55. 115. , Geodesies and curvature in differential geometry in the large, Yeshiva Univ., New York, 1959. 116. G. Reeb, Sur certaines propriétés topologiques des variétés feuilletées, Actualités Sci. Indust. No. 1183, Hermann, Paris, 1952. 117. source
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Among other results Kuiper also found an isometric imbedding of the hyberbolic plane as a closed subset in euclidean three-space.”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
None of the sources cited for this claim have been checked yet.
In the paper
“Without analyticity this is a theorem of Newlander and Nirenberg, whose proof involves delicate considerations in partial differential equations”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“The famous theorem of Picard says that if ƒ is not a constant, the set Pi-f(Ei) cannot contain more than two points.”
§ X = Xi A * X + 7T2 A * 2 , > II. Equidistribution.
What the sources say
- Not matchedcites Picard
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The affirmative answer to this for m = 1 is the classical Picard theorem.”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“Chern and Lashof proved that a compact immersed submanifold in E m has total curvature ^2, and if it is equal to 2, it is a convex hypersurface imbedded in an euclidean space of one dimension higher [44].”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
What the sources say
- Not matchedcites [44] 44. f Q n a generalization of Kâhler geometry, Algebraic geometry and topol- ogy (Lefschetz Jubilee Volume), Princeton Univ. Press, Princeton, N.J., 1957, pp. 103-121.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
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- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- Not matched
None of the sources cited for this claim have been checked yet.
In the paper
“It says that a complete minimal surface in E z is a plane, if its image under the Gauss mapping omits an open subset of the Grassmann manifold (which in this case is the unit sphere).”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“For an irreducible linear group G, Kobayashi and Nagano found a necessary and sufficient condition for a G-structure to be of finite type in terms of their theory of filtered Lie algebras [77].”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
What the sources say
- Not matchedcites [77] 77. f o n filtered Lie algebras and geometnc structures. I, J. Math. Mech. 13 (1964), 875-908;
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“That this implies local flatness is an easy theorem in the real analytic case.”
§ X = Xi A * X + 7T2 A * 2 , > By imbedding
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ X = Xi A * X + 7T2 A * 2 , > Compact complex manifolds.
“He, and later Akizuki and Nakano, gave sufficient conditions of a differential-geometric nature for the vanishing of the cohomology groups H q (M, A P (W)) (for the particular case of a line bundle W) [2], [80 ].”
§ X = Xi A * X + 7T2 A * 2 , > References
“2. A. D. Alexandrow, Die innere Geometrie der konvexen Flâchen, Akademie Verlag, Berlin, 1955.”
Citations
Cites
Foundations of Differential Geometry
Kobayashi, S., & Nomizu, K. (1963) · UCLA - Biblioteca de Ciencias y Tecnologia Felix Morales Bueno · Cited by 6,175
Differential Geometry and Symmetric Spaces
Helgason, S. (2001) · American Mathematical Society eBooks · doi:10.1090/chel/341 · Cited by 3,009
The Topology of Fibre Bundles.
Steenrod, N. E. (1951) · Cited by 2,216
On the foundations of differential geometry
Rubio, R. G. (1964) · Časopis pro pěstování matematiky · doi:10.21136/cpm.1964.117506 · Cited by 2,021
Analytic Functions of Several Complex Variables.
Gray, J. W., Gunning, R. C., & Rossi, H. (1968) · American Mathematical Monthly · doi:10.2307/2313472 · Cited by 1,382
Lectures on Differential Geometry.
Vanstone, J. R., & Sternberg, S. (1966) · American Mathematical Monthly · doi:10.2307/2315376 · Cited by 1,207
Sur La Cohomologie des Espaces Fibres Principaux et des Espaces Homogenes de Groupes de Lie Compacts
Borel, A. (1953) · The Annals of Mathematics · doi:10.2307/1969728 · Cited by 986
Analytic Functions of Several Complex Variables
Gunning, R. C., & Rossi, H. (2009) · American Mathematical Society eBooks · doi:10.1090/chel/368 · Cited by 923
11. The Imbedding Problem for Riemannian Manifolds
Nash, J. (2002) · Princeton University Press eBooks · doi:10.1515/9781400884087-013 · Cited by 838
Complex Analytic Connections in Fibre Bundles
Atiyah, M. F. (1957) · Transactions of the American Mathematical Society · doi:10.1090/s0002-9947-1957-0086359-5 · Cited by 812
Research works printed but not linked: 26
- [6]6. A. Andronov and L. Pontrjagin, Systèmes grossiers, C.R. Acad. Sci. Paris 14 (1937), 247-250.
- [8]8. •, Some examples of complex manifolds, Bonn. Math. Schr. 6 (1958), 1-28.
- [11]11. 1 Charakteristiche Klassen und Anwendungen, Enseignement Math. 7 (1961), 188-213.
- [16]16. Les variétés riemanniennes dont la courbure satisfait certaines condi- tions, pp. 447-456, Proc. Internat. Congress Math., 1962.
- [18]18. L. Bers, Spaces of Riemann surfaces, pp. 349-361, Proc. Internat. Congress Math., 1958.
- [23]23. W. Blaschke and G. Bol, Geometrie der Gewebe, Springer, Berlin, 1938.
- [32]32. f L CS sousgroupes des groupes continus de transformations, Ann. Sci. École Norm Sup. Paris 25 (1908), 57-194 ( = Oeuvres complètes, Partie II, pp. 719- 856).
- [34]34. > s ur i a possibilité de plonger un espace riemannien donné dans un espace euclidien, Ann. Soc. Polon. Math. 6 (1927), 1-7 ( =
- [35]35. Oeuvres complètes, Partie III, pp. 1091-1097).
- [38]38. H. Cartan, Variétés analytiques complexes et cohomologie, Colloque tenu à Bruxelles, 1953, pp. 41-55;
- [42]42. Pseudo-groupes continus infinis. Geometrie différentielle, Colloquium, Strasbourg, pp. 119-136, 1953, Centre National de la Recherche Scientifique, Paris, 1953.
- [46]46. II, Michigan Math. J 5 (1958), 5-12.
- [52]52. Thone, Liège, 1951. 51. 1? $ ur i a théorie des variétés feuilletées, Univ. Roma 1st. Naz. Alta Rend Mat. e Appl. (5) 10 (1951), 64-82.
- [58]58. D. Gromoll, Differenzierbare Strukturen und Metriken positiver Krümmung auf Sphàren, Dissertation, Univ. of Bonn, Bonn, 1964.
- [61]61. Variétés feuilletées, Ann. Scuola Norm. Sup. Pisa. 16 (1962), 367-397.
- [66]66. Komplexe Mannigfalligkeiten, pp. 119-136, Proc. Internat. Congress Math., 1958.
- [70]70. II, Duke Math. J. 8 (1941), 11-46.
- [77]77. f o n filtered Lie algebras and geometnc structures. I, J. Math. Mech. 13 (1964), 875-908;
- [78]78. II, J. Math. Mech. 14 (1965), 513-521;
- [79]79. III, J. Math. Mech. 14 (1965), 679-706;
- [80]80. IV, J. Math. Mech. 15 (1966), 163-175;
- [85]85. On Kdhler varieties of restricted type, Ann. of Math. 60 (1954), 28-48.
- [87]87. Q n deformations of complex analytic structures. I, II, Ann. of Math. 67 (1958), 328-466;
- [88]88. III, Ann. of Math. 71 (1960), 43-76.
- [91]91. II, Indag. Math. 17 (1955), 683-689.
- [98]98. P. Liebermann, Problèmes d'équivalence relatifs à une structure presque com- plexe sur une variété à quatre dimensions, Acad. Roy. Belg. Bull. Cl. Sci. (5) 36 (1950), 742-755.
Cited by
Sorted by citations
On the existence of contact forms
Thurston, W. P. H., & Winkelnkemper, H. E. (1975) · Proceedings of the American Mathematical Society · doi:10.1090/s0002-9939-1975-0375366-7 · Cited by 247
Formes de Contact sur les Variétés de Dimension 3
Martinet, J. (1971) · Lecture notes in mathematics · doi:10.1007/bfb0068901 · Cited by 158
The Calabi–Yau conjectures for embedded surfaces
Colding, T., & Minicozzi, W. P. (2008) · Annals of Mathematics · doi:10.4007/annals.2008.167.211 · Cited by 148
“true. Their original form was given in 1965 in [Ca] where E. Calabi made the following two conjectures about minimal surfaces (they were also promoted by S.S. Chern at the same time; see page 212 of [Ch]): Conjecture 0.1. “Prove that a complete minimal hypersurface in Rn must be unbounded.” Calabi continued: “It is kn…”
ResultInfluentialExistence and classification of overtwisted contact structures in all dimensions
Borman, M. S., Eliashberg, Y., & Murphy, E. (2015) · Acta Mathematica · doi:10.1007/s11511-016-0134-4 · Cited by 134
Complex and quaternionic analyticity in chiral and gauge theories, I
Gürsey, F., & Tze, H. C. (1980) · Annals of Physics · doi:10.1016/0003-4916(80)90056-1 · Cited by 128
Metadata
- Authors
- Chern, S. S.
- Corresponding author
- Chern, S. S.
- Venue
- Bulletin of the American Mathematical Society
- Publisher
- American Mathematical Society (AMS)
- Published
- 1966-03-01
- Volume
- 72
- Issue
- 2
- Pages
- 167-220
- ISSN
- 0002-9904 (ISSN-L)
- OpenAlex
- W1980777192
- Cited by
- 159
- References
- 143
- Citations by year
- 2026: 3 · 2025: 3 · 2024: 3 · 2023: 2 · 2022: 2 · 2021: 7 · 2020: 7 · 2019: 2 · 2018: 4 · 2016: 4 · 2015: 6 · 2014: 3 · 2013: 4 · 2012: 5
- FWCI
- 2.81
- Type
- journal article
- Language
- en
- Open access
- diamond
- doi.org
- Locations
- Bulletin of the American Mathematical Society (publishedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Geometry and Topology › Advanced Differential Geometry Research
- Physical Sciences › Mathematics › Geometry and Topology › Geometry and complex manifolds
- Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Keywords
- G-structures, geometric structures on manifolds, Riemannian structure, complex structure, vector fields, Euler characteristic, harmonic forms, parallel vector fields
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