The geometry of G-structures

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],”

What the sources say

• ?Could 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)

  cites [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
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  cites [39] 39. Thone, Liège, 1953.

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].”

What the sources say

• ?Could 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)

  cites [53] 53. T. Frankel, Manifolds with positive curvature, Pacific J. Math. 11 (1961), 165- 174. sourceExperimental — accuracy not yet measured

In the paper

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

In the paper

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

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.”

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.”

What the sources say

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  cites [132]

In the paper

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

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.”

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,”

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.”

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.”

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  cites [l40]

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.”

In the paper

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

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.

In the paper

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

In the paper

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

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.”

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

In the paper

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

In the paper

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

In the paper

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

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

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.”

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.”

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.”

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].”

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.

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.”

What the sources say

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  cites [112]

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).”

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.”

In the paper

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

What the sources say

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  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.

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.”

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.”

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.”

What the sources say

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  cites [69] 69. W. Kaplan, Regular curve-families filling the plane, I, Duke Math. J. 7 (1940), 154-185;

In the paper

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

In the paper

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

In the paper

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

In the paper

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

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].”

What the sources say

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  cites [116]

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.”

What the sources say

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  cites Denjoy

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

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  cites [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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  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.
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  cites [135]

In the paper

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

What the sources say

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  cites Poincaré-Bendixon

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.”

What the sources say

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  cites Denjoy
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  cites Schwartz

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.”

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 ].”

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.

In the paper

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

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].”

What the sources say

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  cites [23] 23. W. Blaschke and G. Bol, Geometrie der Gewebe, Springer, Berlin, 1938.

In the paper

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

What the sources say

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  cites [6] 6. A. Andronov and L. Pontrjagin, Systèmes grossiers, C.R. Acad. Sci. Paris 14 (1937), 247-250.

In the paper

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

What the sources say

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  cites Riemann

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].”

What the sources say

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  cites [67] 67. F. Hirzebruch and K. Kodaira, On the complex projective spaces, J. Math. Pures Appl. 36 (1957), 201-216.

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.”

What the sources say

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  cites Lima and H. Rosenberg

In the paper

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

In the paper

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

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”

What the sources say

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  cites Kuranishi

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.”

In the paper

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

What the sources say

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  cites Chow

In the paper

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

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.
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  cites [124]

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].”

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

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.”

What the sources say

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  cites Smale

In the paper

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

What the sources say

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  cites [95] 95. 1 Introduction to differentiable manifolds, Interscience, New York, 1962.

In the paper

“Let”

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.”

What the sources say

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  cites [113]

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.”

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].”

What the sources say

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  cites [133]

In the paper

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

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.”

In the paper

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

In the paper

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

What the sources say

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  cites Atiyah-Singer

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.”

In the paper

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

In the paper

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

What the sources say

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  cites [122]

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.”

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.”

What the sources say

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  cites [122]

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 ].”

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

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.)”

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.”

What the sources say

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  cites Cartan and Serre

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.”

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.”

What the sources say

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  cites Kodaira and Spencer

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 ].”

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
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  cites [80] 80. IV, J. Math. Mech. 15 (1966), 163-175;

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.”

What the sources say

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  cites [81] 81. V, (to appear).

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.”

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.”

What the sources say

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  cites [111]

In the paper

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

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].”

What the sources say

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  cites [85] 85. On Kdhler varieties of restricted type, Ann. of Math. 60 (1954), 28-48.
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  cites [99] 99. A. Lichnerowicz, Théorie globale des connexions et des groupes d'holonomie, Edizioni Cremonese, Rome, 1957.

In the paper

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

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.”

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.”

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.”

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].”

What the sources say

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  cites [35] 35. Oeuvres complètes, Partie III, pp. 1091-1097).

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].”

What the sources say

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  cites [89] 89. V. Yoh Kraines, Topology of quaternionic manifolds, Bull. Amer. Math. Soc. 71 (1965), 526-527.
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  cites [97] 97. H. Le vine, A theorem on holomorphic mappings into complex projective space, Ann. of Math. 71 (1960), 529-535.
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  cites [lOO]

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.”

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.”

In the paper

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

What the sources say

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  cites Nevanlinna

In the paper

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

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

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 ].”

What the sources say

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  cites [131]

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.”

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.”

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.”

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

In the paper

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

In the paper

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

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.”

What the sources say

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  cites Picard

In the paper

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

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].”

What the sources say

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  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.

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).”

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].”

What the sources say

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  cites [77] 77. f o n filtered Lie algebras and geometnc structures. I, J. Math. Mech. 13 (1964), 875-908;

In the paper

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