The geometry of G-structures

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The Euler-Poincaré characteristic of a compact orientable riemannian manifold M of even dimension n = 2m is given by the Gauss-Bonnet formula.

“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],”

• cites [4] 4. W. Ambrose and I. M. Singer, A theorem on holonomy, Trans. Amer. Math. Soc. 75 (1953), 428-443. sourceCould not verify

  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)

  Experimental — accuracy not yet measured
• cites [39] 39. Thone, Liège, 1953.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Frölicher and Nijenhuis proved that the complex structure on a compact complex manifold M is rigid if H^1(M,Θ)=0, where H^1(M,Θ) is the one-dimensional cohomology group of M over the sheaf of germs of holomorphic vector fields.

“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].”

• cites [53] 53. T. Frankel, Manifolds with positive curvature, Pacific J. Math. 11 (1961), 165- 174. sourceCould not verify

  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)

  Experimental — accuracy not yet measured

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.

In Example 3, λ is the gaussian curvature of the riemannian structure.

“One sees that X is the gaussian curvature of the riemannian structure.”

The quote only partly states this claim.

For a holonomic G-structure on M, if Q is multiplication by an invariant exterior differential form, then Q commutes with the Laplace-de Rham operator: QΔ=ΔQ.

“Moreover, if Q is the multiplication by an invariant exterior differential form, we have also ÇA=A<2.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The complex projective space with the usual elliptic hermitian metric has sectional curvatures R satisfying A/4 ≤ R ≤ A, where A is a positive constant.

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

According to Wang, the compact symmetric riemannian manifolds of rank 1, together with the spherical and elliptic spaces, are the only compact and connected two-point homogeneous spaces.

“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.”

• cites [132]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

Parallelism is at the basis of the notion of curvature; geometrically curvature measures the dependence of parallelism on the curve.

“Parallelism is at the basis of the notion of curvature, for geometrically curvature measures the dependence of parallelism on the curve.”

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.

In high dimensions, constancy of the sectional curvature leads to the space forms, and not every manifold has a riemannian metric of constant curvature.

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Rauch's theorem: A complete simply connected a-pinched riemannian manifold with a = 0.75 is homeomorphic to the sphere.

“Rauch proved the following theorem : A complete simply connected a-pinched riemannian manifold with a = 0.75 is homeomorphic to the sphere,”

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

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.

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Yamabe theorem: A compact riemannian manifold of dimension ≥ 3 is conformally equivalent to one of constant scalar curvature.

“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.”

• cites [l40]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper may give no reason why it holds.

An Einstein metric on a three-dimensional manifold is necessarily of constant curvature.

“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.”

It may state two separate claims as one.

The conjecture on signs of Euler characteristic under sectional curvature is true for n = 4 and for the case that M has constant sectional curvature.

“The statement is true for w = 4 [42] and for the case that M has constant sectional curvature.”

• cites [42] 42. Pseudo-groupes continus infinis. Geometrie différentielle, Colloquium, Strasbourg, pp. 119-136, 1953, Centre National de la Recherche Scientifique, Paris, 1953.Not checked yet

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

An O(n)-structure is nothing else but a riemannian structure.

“An 0(n)-structure is nothing else but a riemannian structure.”

It may state two separate claims as one.

If a complete riemannian manifold has Ricci curvature equal to a positive constant, then its universal covering manifold is compact and its fundamental group is finite.

“It follows that its universal covering manifold is also compact and hence that its fundamental group is finite.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Gromoll-Calabi diffeomorphism theorem: There exists a number δ_n depending on n (0 < δ_n < 1, lim δ_n = 1) such that if a simply connected riemannian manifold M of dimension n is δ_n-pinched, then M is diffeomorphic to the n-sphere with its usual differentiable structure.

“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.”

• cites [57] 57. M. Goto, On algebraic homogeneous spaces, Amer. J. Math. 76 (1954), 811-818. sourceNot checked — no full text of the source

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

The classical example of a connection is the parallelism of Levi-Civita in riemannian geometry.

“The classical example of a connection is the parallelism of Levi-Civita in riemannian geometry.”

It may state two separate claims as one.

The curvature forms define the local properties of a connection and the holonomy groups its global property.

“The curvature forms define the local properties of a connection and the holonomy groups iJ& its global property.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Ambrose and Singer showed how the Lie algebra of the holonomy group can be determined from the curvature forms.

“Ambrose and Singer showed how the Lie algebra of H& can be determined from the curvature forms [5].”

• cites [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. sourceNot checked — no full text of the source

Probably not a substantial claim. The paper’s conclusions may not depend on it.

For n = 2m, a GL(m,C)-structure is what is called an almost complex structure.

“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.”

Probably not a substantial claim.

Synge's lemma: 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 geodesics of M. Then there is a family of closed curves converging toward g, which are all shorter than g.

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

For a compact two-dimensional manifold, the simplest riemannian metric is one of constant gaussian curvature; the curvature is positive for the sphere, zero for the torus, and negative for a surface of genus > 1.

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Andreotti-Frankel theorem: If a compact Kähler manifold of complex dimension 2 has positive sectional curvature, it is complex analytically equivalent to the complex projective plane.

“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].”

• cites [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.Not checked yet

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

A compact riemannian manifold with sectional curvature < 0 has a fundamental group which cannot be abelian and of which every abelian subgroup is cyclic.

“A compact riemannian manifold with sectional curvature <0 has a fundamental group which cannot be abelian and of which every abelian subgroup is cyclic.”

• cites [112]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

It may state two separate claims as one.

Bochner's method applies to the high-dimensional Betti numbers of riemannian manifolds and to the derivation of criteria for the vanishing of more general cohomology groups.

“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).”

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.

Example 7: G is the group of all matrices of order n ≥ 3 of the form λA, λ > 0, where A is orthogonal. This is the conformal structure, to which a normal conformal connection, uniquely determined, can be attached.

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The problem of equivalence for almost complex structure G = GL(m,C), n = 2m, was solved by Libermann for m = 2.

“The problem of equivalence was solved by Libermann for m = 2 [93].”

• cites [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.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

It may state two separate claims as one.

In Example 5, the structure is given by (θ1)^2 + (θ2)^2, determined up to a positive factor; it is the two-dimensional conformal structure.

“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.”

Probably not a substantial claim. It may state two separate claims as one.

A G-structure on M defines an orientation of M and a Riemannian structure, called the associated Riemannian structure.

“Then a G-structure on M defines an orientation of M and a riemannian structure, to be called the associated riemannian structure.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Kaplan proved that to a leaved structure in the plane there exists a continuous real-valued function in the plane, which has neither a maximum nor a minimum and which is constant on the leaves.

“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.”

• cites [69] 69. W. Kaplan, Regular curve-families filling the plane, I, Duke Math. J. 7 (1940), 154-185;Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim.

For G=SO(p)×SO(n-p), a holonomic G-structure means that the manifold has a field of p-dimensional plane elements parallel with respect to the Levi-Civita connection.

“This means that the manifold has a field of ^-dimensional plane elements parallel with respect to the Levi-Civita connection.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

For G=U(m), m=n/2, a holonomic G-structure is essentially the Kähler structure on a complex manifold in its real formulation.

“This is essentially the Kàhler structure on a complex manifold in its real formulation.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Example 6: G = O(n) is the riemannian structure where the Levi-Civita connection can be attached.

“EXAMPLE 6. G -0(n).This is the riemannian structure where the Levi-Civita connection can be attached.”

It may state two separate claims as one.

The two alternatives of the Cartan-Kuranishi theorem correspond respectively to the cases that the two G-structures are locally inequivalent or equivalent.

“They correspond respectively to the cases that the two G-structures are locally inequivalent or equivalent.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Reeb gave a notable example of a leaved structure of codimension one on the three-sphere S^3.

“A notable example of a leaved structure of codimension one on the three-sphere S 3 was given by G. Reeb [116].”

• cites [116]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The fundamental theorem of Denjoy says that, under sufficient smoothness hypotheses, if there is no closed leaf, then every leaf is ergodic.

“The fundamental theorem of Denjoy says that (under sufficient smoothness hypotheses) if there is no closed leaf, then every leaf is ergodic.”

• cites DenjoyNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. It may state two separate claims as one.

A G-structure is holonomic if it satisfies one of three equivalent conditions: the Levi-Civita holonomy group is G or a subgroup of G; permissible frames remain permissible under Levi-Civita parallelism; or there is a connection with structural group NG whose torsion form is zero.

“Such a G-structure is called holonomic, if it satisfies one of the following three conditions, which are equivalent [43], [84], [135]:”

• cites [43] 43. f On curvature and characteristic classes of a Riemann manifold, Abh. Math. Sem. Univ. Hamburg 20 (1955), 117-126.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [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.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [135]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it.

A consequence of the Poincaré-Bendixon theory is that the leaves of a first-order differential equation in the plane go to infinity in both directions.

“A consequence of the Poincaré-Bendixon theory says that the leaves go to infinity in both directions.”

• cites Poincaré-BendixonNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

The Denjoy-Schwartz theorem says that if a compact connected two-dimensional surface is under the action of the real line, then a minimal set is either a single point, a closed curve, or the whole surface, the last case happening only for the torus.

“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.”

• cites DenjoyNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites SchwartzNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

It may state two separate claims as one.

No connection can be introduced depending on this conformal structure alone; otherwise the local homeomorphisms leaving the conformal structure invariant would depend on a finite number of constants.

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

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.

“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 ].”

• cites [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.Not checked yet

It may state two separate claims as one.

S^2×S^2 is a simply connected manifold which admits an infinite number of complex structures.

“Thus S 2 XS 2 is a simply connected manifold which admits an infinite number of complex structures.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Blanchard proved that if a compact Kähler manifold is blown up along a nonsingular submanifold, the resulting manifold is again Kählerian.

“Blanchard proved that if a compact Kâhler manifold is blown up along a nonsingular submanifold, the resulting manifold is again Kâhlerian [23].”

• cites [23] 23. W. Blaschke and G. Bol, Geometrie der Gewebe, Springer, Berlin, 1938.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Andreotti proved that there are no other algebraic structures on S^2×S^2.

“Andreotti proved that there are no other algebraic structures [6].”

• cites [6] 6. A. Andronov and L. Pontrjagin, Systèmes grossiers, C.R. Acad. Sci. Paris 14 (1937), 247-250.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Riemann's mapping theorem says that S^2 has a uniquely determined complex structure.

“Riemann's mapping theorem says that S 2 has a uniquely determined complex structure.”

• cites RiemannNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

Andreotti proved that there is only one Kähler structure on P^2 (the standard one); Hirzebruch and Kodaira proved that there is only one Kähler structure on P^m for odd m.

“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].”

• cites [67] 67. F. Hirzebruch and K. Kodaira, On the complex projective spaces, J. Math. Pures Appl. 36 (1957), 201-216.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Lima and H. Rosenberg generalized this to the result that the rank of a compact simply connected manifold of dimension n is at most n-2.

“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.”

• cites Lima and H. RosenbergNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

If a manifold has a leaved structure of dimension p, it must have a G-structure with G=GL(p,R).

“If a manifold has a leaved structure of dimension p, it must have a G-structure, with G = GL(p, R).”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

H^0(M,S) is the group of all global sections of S.

“Hence H°(M, S) is the group of all global sections of S.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Kuranishi's theorem says that for any compact complex manifold, there exists a universal family of deformations.

“The crowning achievement is the theorem of M. Kuranishi, which says that, for any compact complex manifold, there exists a universal family of deformations”

• cites KuranishiNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

It may state two separate claims as one.

For m>1, a Hopf manifold cannot be given a Kähler structure and is therefore not algebraic, because the latter must have its second Betti number at least 1.

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

By a theorem of Chow, compact submanifolds of P^m are the same as nonsingular algebraic varieties.

“Compact submanifolds of P m are, by a theorem of Chow, the same as the nonsingular algebraic varieties.”

• cites ChowNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

Goto, Borel, and Weil proved that G/T is an algebraic variety and Goto proved that it is rational.

“Goto, Borel, and Weil proved that it is an algebraic variety and Goto proved that it is rational [56], [124].”

• cites [56] 56. S. I. Goldberg, Curvature and homology, Academic Press, New York, 1962.Not checked — no full text of the source
• cites [124]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it.

If G is a connected compact Lie group and T a maximal toroid of G, Borel observed that G/T has a complex structure.

“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].”

• cites [25] 25. A. Borel and J.-P. Serre, Groupes de Lie et puissances réduits de Steenrod, Amer. J. Math. 75 (1953), 409-448. sourceNot checked — no full text of the source

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Smale gave examples of manifolds in which the structurally stable vector fields are not dense in the set of all vector fields.

“In fact, Smale gave examples of manifolds in which the structurally stable vector fields are not dense in the set of all vector fields.”

• cites SmaleNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Lima proved that S^3 has rank one.

“Lima proved that S z has rank one [95].”

• cites [95] 95. 1 Introduction to differentiable manifolds, Interscience, New York, 1962.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

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.

For an exact sequence of sheaves 0→R→S→T→0 on M, the sequence of cohomology groups 0→H^0(M,R)→H^0(M,S)→H^0(M,T)→H^1(M,R)→H^1(M,S)→... is exact.

“Let”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Pugh's closing lemma implies that the closure of the set of all nonwandering points is the closure of the set of all closed trajectories.

“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.”

• cites [113]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it.

If S is the sheaf of germs of mappings into a topological group Y (not necessarily abelian), H^1(M,S) is the set of all equivalence classes (in the sense of bundles) of bundles with the group Y.

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

Wang determined all the compact homogeneous complex manifolds with a finite fundamental group, among which are the simply connected even-dimensional compact Lie groups.

“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].”

• cites [133]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it.

A Hopf manifold is topologically homeomorphic to S^1×S^{2m-1}.

“Topologically it is homeomorphic to S 1 XS 2m ~l.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

A homomorphism of sheaves S→S' induces a homomorphism H^q(M,S)→H^q(M,S') of the corresponding cohomology groups.

“A homomorphism of sheaves S-+S' induces a homomorphism H q (M, S)^>H q (M, S') of the corresponding cohomology groups.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

An orientable manifold of two real dimensions always has a complex structure.

“An orientable manifold of two (real) dimensions always has a complex structure.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The Atiyah-Singer theorem says that the topological index i_t(D) equals the analytical index i_a(D).

“The Atiyah-Singer theorem says that it{D) =i a (D).”

• cites Atiyah-SingerNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The second Cousin problem has a solution on a Stein manifold if H^2(M,Z)=0.

“(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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

Every meromorphic function on a Stein manifold is the quotient of two holomorphic functions.

“(3) Every meromorphic function on a Stein manifold is the quotient of two holomorphic functions.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Serre's duality theorem is valid for noncompact complex manifolds.

“Actually Serre's duality theorem is valid for noncompact complex manifolds; we state it for compact manifolds for simplicity.”

• cites [122]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The classical Riemann-Roch theorem is l(D)=d(D)-g+i+1, where g is the genus of M and i>=0 is the index of specialty of D.

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

Serre's duality theorem says that H^q(M, A^r(W)) and H^{m-q}(M, A^{m-r}(W*)) are in duality and hence have the same dimension.

“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.”

• cites [122]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

It may state two separate claims as one.

Weil's theorem states that the cohomology class determined by F(Φ) is independent of the choice of connection and depends only on the bundle.

“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 ].”

• cites [40] 40. S. S. Chern, On the curvatura intégra in a riemannian manifold, Ann. of Math. 46 (1945), 674-684. sourceNot checked — no full text of the source

It may state two separate claims as one.

For G=SO(q) and G'=SO(q-1), a necessary condition for reduction of the G-bundle to a G'-bundle is that the Euler class be zero, and in this case the condition is also sufficient.

“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.)”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

For compact M, H^q(M,S) is finite-dimensional for any analytic coherent sheaf S.

“When M is compact, Cartan and Serre proved that H q (M, S) is of finite dimension for any analytic coherent sheaf 5.”

• cites Cartan and SerreNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The Riemann-Roch theorem can be written as dim H^0(M,Ω(D)) - dim H^1(M,Ω(D)) = d - g + 1.

“Then the Riemann-Roch theorem can be written (54) dim H°(M, Q(D)) -dim E l (M, ti(D)) = d -g + 1.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Kodaira and Spencer proved that every line bundle over a nonsingular algebraic variety is defined by a divisor.

“Conversely, it was proved by Kodaira and Spencer that every line bundle over a nonsingular algebraic variety is defined by a divisor.”

• cites Kodaira and SpencerNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Kodaira, and later Akizuki and Nakano, gave sufficient differential-geometric conditions for the vanishing of H^q(M, A^p(W)) for a line bundle W.

“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 ].”

• cites [2] 2. A. D. Alexandrow, Die innere Geometrie der konvexen Flâchen, Akademie Verlag, Berlin, 1955. sourceNot checked — no full text of the source
• cites [80] 80. IV, J. Math. Mech. 15 (1966), 163-175;Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Kodaira's theorem says that if V is a Kähler manifold whose fundamental two-form has rational periods, then there exists a one-to-one holomorphic mapping f: V->P_m.

“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.”

• cites [81] 81. V, (to appear).Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The Riemann-Hurwitz formula states that for Riemann surfaces V and M and a holomorphic mapping f: V->M of degree d, χ(V)+w = d χ(M), where w is the index of ramification.

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Porteous derived relations between characteristic classes for dilatations and for ramified coverings with singularities of a relatively simple type.

“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.”

• cites [111]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

A complete minimal submanifold in E^m is never compact.

“A complete minimal submanifold in E m is never compact.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

A C^1-riemannian manifold V of dimension n can be C^1 isometrically imbedded in an euclidean space of dimension 2n+1 (Nash's theorem, improved by Kuiper).

“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].”

• cites [85] 85. On Kdhler varieties of restricted type, Ann. of Math. 60 (1954), 28-48.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [99] 99. A. Lichnerowicz, Théorie globale des connexions et des groupes d'holonomie, Edizioni Cremonese, Rome, 1957.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

Cartan's determination of the simple infinite pseudo-groups (over the complex field) has not been completely verified.

“Cartan's determination of the simple infinite pseudo-groups (over the complex field) has not been completely verified”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Nash proved that a compact two-dimensional riemannian manifold can be isometrically imbedded in an euclidean space of dimension 17.

“For example, Nash proved that a compact two-dimensional riemannian manifold can be isometrically imbedded in an euclidean space of dimension 17.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

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.

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Two tightly imbedded analytic compact surfaces of genus one in euclidean three-space E^3 differ by an isometry of E^3, if they are isometric (A. D. Alexandrow).

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

If V and M are real analytic riemannian manifolds of dimensions n and m = n(n+1)/2 respectively, then there is locally an isometric mapping of V into M (the theorem of Janet-Cartan).

“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].”

• cites [35] 35. Oeuvres complètes, Partie III, pp. 1091-1097).Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

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.

“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].”

• cites [89] 89. V. Yoh Kraines, Topology of quaternionic manifolds, Bull. Amer. Math. Soc. 71 (1965), 526-527.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [97] 97. H. Le vine, A theorem on holomorphic mappings into complex projective space, Ann. of Math. 71 (1960), 529-535.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [lOO]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

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 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.

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

The second fundamental form is related to the curvature forms of both riemannian manifolds by the (generalized) Gauss equations.

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Nevanlinna obtained sharper results generalizing Picard's theorem in the form of his defect relations.

“Much sharper results generalizing Picard's theorem were obtained by R. Nevanlinna in the form of his defect relations.”

• cites NevanlinnaNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it.

From the Bott-Chern study of differential forms in the vector bundle an equidistribution theorem was derived.

“From it an equidistribution theorem was derived, but much remains to be done.”

• cites [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. sourceNot checked — no full text of the source

Probably not a substantial claim. The paper’s conclusions may not depend on it.

A closed convex surface (Gaussian curvature >= 0) in euclidean three-space E^3 is completely determined by its metric, modulo the isometries of E^3 (a classical theorem of Cohn-Vossen, complemented by a remark of K. Voss).

“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 ].”

• cites [131]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

It may state two separate claims as one.

For a given compact manifold the minimum of the total curvatures for all possible immersions is a differential topological invariant and is an integer.

“For a given compact manifold the minimum of the total curvatures for all possible immersions is a differential topological invariant and is an integer.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

The classical example of Fatou and Bieberbach is a holomorphic mapping f: E^2 -> E^2 whose Jacobian determinant is identically equal to one, while E^2 - f(E^2) contains an open subset.

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

To remove the assumption of analyticity Nirenberg had to put in some additional hypotheses, so that the question whether the analyticity assumption in Alexandrow's theorem could be removed remains unsettled.

“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.”

• cites [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. sourceNot checked — no full text of the source

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Kuiper found an isometric imbedding of the hyperbolic plane as a closed subset in euclidean three-space.

“Among other results Kuiper also found an isometric imbedding of the hyberbolic plane as a closed subset in euclidean three-space.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Without analyticity, that the vanishing of the structure function implies local flatness is a theorem of Newlander and Nirenberg.

“Without analyticity this is a theorem of Newlander and Nirenberg, whose proof involves delicate considerations in partial differential equations”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Picard's theorem says that if f is not constant, the set P_1 - f(E^1) cannot contain more than two points.

“The famous theorem of Picard says that if ƒ is not a constant, the set Pi-f(Ei) cannot contain more than two points.”

• cites PicardNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it.

For m = 1, the question of whether one of 2m+1 linear subspaces of dimension m-1 of P^{2m-1} in general position intersects f(C) has an affirmative answer, which is the classical Picard theorem.

“The affirmative answer to this for m = 1 is the classical Picard theorem.”

It may state two separate claims as one.

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 (Chern and Lashof).

“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].”

• cites [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.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it.

A complete minimal surface in E^3 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) (a classical theorem of S. Bernstein, as generalized by R. Osserman).

“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).”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

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.

“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].”

• cites [77] 77. f o n filtered Lie algebras and geometnc structures. I, J. Math. Mech. 13 (1964), 875-908;Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim. The paper’s conclusions may not depend on it.

That the vanishing of the structure function implies local flatness is an easy theorem in the real analytic case.

“That this implies local flatness is an easy theorem in the real analytic case.”