Topic
Tangent bundle
About: Tangent bundle is a research topic. Over the lifetime, 3038 publications have been published within this topic receiving 42176 citations.
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Book•
13 Oct 2011
TL;DR: Weinstein and Sturmouasville as discussed by the authors introduced the concept of projective spaces as base spaces of the Hopf Fibrations and showed that they can be used to define the topology of the Cayley Projective Plane.
Abstract: 0. Introduction.- A. Motivation and History.- B. Organization and Contents.- C. What is New in this Book?.- D. What are the Main Problems Today?.- 1. Basic Facts about the Geodesic Flow.- A. Summary.- B. Generalities on Vector Bundles.- C. The Cotangent Bundle.- D. The Double Tangent Bundle.- E. Riemannian Metrics.- F. Calculus of Variations.- G. The Geodesic Flow.- H. Connectors.- I. Covariant Derivatives.- J. Jacobi Fields.- K. Riemannian Geometry of the Tangent Bundle.- L. Formulas for the First and Second Variations of the Length of Curves.- M. Canonical Measures of Riemannian Manifolds.- 2. The Manifold of Geodesics.- A. Summary.- B. The Manifold of Geodesics.- C. The Manifold of Geodesics as a Symplectic Manifold.- D. The Manifold of Geodesics as a Riemannian Manifold.- 3. Compact Symmetric Spaces of Rank one From a Geometric Point of View.- A. Introduction.- B. The Projective Spaces as Base Spaces of the Hopf Fibrations.- C. The Projective Spaces as Symmetric Spaces.- D. The Hereditary Properties of Projective Spaces.- E. The Geodesics of Projective Spaces.- F. The Topology of Projective Spaces.- G. The Cayley Projective Plane.- 4. Some Examples of C- and P-Manifolds: Zoll and Tannery Surfaces.- A. Introduction.- B. Characterization of P-Metrics of Revolution on S2.- C. Tannery Surfaces and Zoll Surfaces Isometrically Embedded in (IR3, can).- D. Geodesics on Zoll Surfaces of Revolution.- E. Higher Dimensional Analogues of Zoll metrics on S2.- F. On Conformal Deformations of P-Manifolds: A. Weinstein's Result.- G. The Radon Transform on (S2, can).- H. V. Guillemin's Proof of Funk's Claim.- 5. Blaschke Manifolds and Blaschke's Conjecture.- A. Summary.- B. Metric Properties of a Riemannian Manifold.- C. The Allamigeon-Warner Theorem.- D. Pointed Blaschke Manifolds and Blaschke Manifolds.- E. Some Properties of Blaschke Manifolds.- F. Blaschke's Conjecture.- G. The Kahler Case.- H. An Infinitesimal Blaschke Conjecture.- 6. Harmonic Manifolds.- A. Introduction.- B. Various Definitions, Equivalences.- C. Infinitesimally Harmonic Manifolds, Curvature Conditions.- D. Implications of Curvature Conditions.- E. Harmonic Manifolds of Dimension 4.- F. Globally Harmonic Manifolds: Allamigeon's Theorem.- G. Strongly Harmonic Manifolds.- 7. On the Topology of SC- and P-Manifolds.- A. Introduction4.- B. Definitions.- C. Examples and Counter-Examples.- D. Bott-Samelson Theorem (C-Manifolds).- E. P-Manifolds.- F. Homogeneous SC-Manifolds.- G. Questions.- H. Historical Note.- 8. The Spectrum of P-Manifolds.- A. Summary.- B. Introduction.- C. Wave Front Sets and Sobolev Spaces.- D. Harmonic Analysis on Riemannian Manifolds.- E. Propagation of Singularities.- F. Proof of the Theorem 8. 9 (J. Duistermaat and V. Guillemin).- G. A. Weinstein's result.- H. On the First Eigenvalue ?1=?12.- Appendix A. Foliations by Geodesic Circles.- I. A. W. Wadsley's Theorem.- II. Foliations With All Leaves Compact.- Appendix B. Sturm-Liouville Equations all of whose Solutions are Periodic after F. Neuman.- I. Summary.- II. Periodic Geodesics and the Sturm-Liouville Equation.- III. Sturm-Liouville Equations all of whose Solutions are Periodic.- IV. Back to Geometry with Some Examples and Remarks.- Appendix C. Examples of Pointed Blaschke Manifolds.- I. Introduction.- II. A. Weinstein's Construction.- III. Some Applications.- Appendix D. Blaschke's Conjecture for Spheres.- I. Results.- II. Some Lemmas.- III. Proof of Theorem D.4.- Appendix E. An Inequality Arising in Geometry.- Notation Index.
1,042 citations
700 citations
TL;DR: In this article, the authors consider the problem of finding an isomorphism in a set of subsets of a TM and show that there exists a neighborhood W 1, W 2, W 3 of (p, Xp), (p); F ( Xp) and F (Xp) respectively such that W 1 is an open set.
Abstract: of subsets of TM: Note that i) 8 (p;Xp) 2 TM , as p 2M ) there exists (U ; ) 2 S such that p 2 U ; i.e. (p;Xp) 2 TU , and we have TU = 1 (R) 2 : ii) If we de
ne F : TpM ! R by F (Xp) = (Xp(x); Xp(x); :::::; Xp(x)) where x; x; ::::; x are local coordinates on (U ; ), then clearly F is an isomorphism, so (p; Xp) = ( (p); F ( Xp)); and 1 = ( 1 ; F 1 ): Now take 1 (U); 1 (V ) 2 and suppose (p; Xp) 2 1 (U)\ 1 (V ) for some U; V open in R: ) (p; Xp) 2 U and (p; Xp) 2 V Take U = U U and V = V V where U ; U ; V ; V are open sets in R:Clearly (p) 2 U ; F ( Xp) 2 U ; (p) 2 V , and F (Xp) 2 V : From the de
nition of open set there exist neighborhoodsW 1 ;W 2 ;W 1 ;W 2 of (p); (p); F ( Xp) and F (Xp) respectively such that
695 citations
1 Jan 2007
TL;DR: In this article, it was shown that the weak limit of a sequence of smooth closed real (1, 1)-currents with small negative part can be bounded in terms of the Lelong numbers of T, once a lower bound for the curvature of the tangent bundle TX is known.
Abstract: — LetX be a compact complex manifold and let T be a closed positive current of bidegree (1, 1) on X . It is shown that T is the weak limit of a sequence (Tk) of smooth closed real (1, 1)-currents with small negative part. The negative part of the Tk ’s can be bounded in terms of the Lelong numbers of T , once a lower bound for the curvature of the tangent bundle TX is known. Moreover, Kiselman’s procedure for killing Lelong numbers of a plurisubharmonic function is extended to manifolds by an alternative method based on Hormander’s L estimates for ∂. These results are then applied to derive various results concerning divisors or intersection theory in the context of analytic geometry. Especially, we obtain a relation between effective and numerically effective divisors on arbitrary compact manifolds, and we show that every manifold X in the Fujiki class C with nef tangent bundle is Kahler. If D is an effective divisor in a Kahler manifold, we also obtain a general self-intersection inequality giving a bound of the degrees of the constant multiplicity strata of D, in terms of a polynomial in the cohomology class {D} ∈ H(X, IR).
587 citations
TL;DR: In this paper, it is shown that if the manifold is orientable then this property is equivalent to the existence of a globally defined 1-form ca of maximal rank, and then some further equivalent conditions are derived.
Abstract: transformations in general and to the study of global contact transformations in the special case of euclidean space. In attempting to generalize Lie's results to more general manifolds, it becomes clear that there are intrinsic global differences between the even and odd dimensional cases. In this paper, only the odd dimensional case will be discussed. Intuitively, a manifold carries a contact structure if the coordinate transformations can be chosen to preserve the 1-form dz - y'dx' up to a non-zero, multiplicative factor. We first show that if the manifold is orientable then this property is equivalent to the existence of a globally defined 1form ca of maximal rank, and then we derive some further equivalent conditions. It is well known that the existence of such a 1-form implies that the structure group of the tangent bundle can be reduced to the unitary group. (See, e.g., Chern, [7]). If this can be done, we say that the manifold is an almost-contact manifold. The obstructions to the existence of such a structure are investigated and it is shown that the primary obstruction is the third Stiefel-Whitney class. This solves completely the question of the existence of U(2) structures on five dimensional manifolds. We turn then to a discussion of global contact transformations, i.e., transformations which preserve ca up to a non-zero, multiplicative factor, T. Lie's results are shown to be valid in general by providing intrinsic proofs of his theorems. It should be noted that in this context, in general, analysis occurs only in the definitions, while the proofs consist simply of algebraic manipulations. Sheaves are employed at this point only because they provide a convenient language. Finally, we show that the factors T which can occur in contact transformations are not arbitrary. These results are then applied to the study of deformations (in the
440 citations
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