By B.A. Dubrovin
Up till lately, Riemannian geometry and easy topology weren't incorporated, even by means of departments or colleges of arithmetic, as obligatory topics in a university-level mathematical schooling. the normal classes within the classical differential geometry of curves and surfaces which have been given in its place (and nonetheless are given in a few areas) have come progressively to be seen as anachronisms. notwithstanding, there was hitherto no unanimous contract as to precisely how such classes will be cited so far, that's to assert, which elements of recent geometry may be considered as totally necessary to a latest mathematical schooling, and what will be definitely the right point of abstractness in their exposition. the duty of designing a modernized path in geometry used to be began in 1971 within the mechanics department of the school of Mechanics and arithmetic of Moscow nation college. The subject-matter and point of abstractness of its exposition have been dictated by means of the view that, as well as the geometry of curves and surfaces, the next subject matters are definitely necessary within the a number of parts of software of arithmetic (especially in elasticity and relativity, to call yet two), and are for this reason crucial: the speculation of tensors (including covariant differentiation of them); Riemannian curvature; geodesics and the calculus of diversifications (including the conservation legislation and Hamiltonian formalism); the actual case of skew-symmetric tensors (i. e.
Read or Download Modern Geometry Methods and Applications: Part II: The Geometry and Topology of Manifolds (Graduate Texts in Mathematics) (Part 2) PDF
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Extra info for Modern Geometry Methods and Applications: Part II: The Geometry and Topology of Manifolds (Graduate Texts in Mathematics) (Part 2)
V. Pogorelov, Ju. F. Borisov, V. A. Toponogov and V. I. Kuz'minov, who during reviewing the e-book contributed many helpful reviews. We additionally thank Ja. B. Zel'dovie for numerous observations resulting in advancements within the exposition at numerous issues, in connexion with the instruction of the English and French versions of this e-book. We provide our certain thank you additionally to the students who facilitated the duty of incorporating the fewer general fabric into the e-book. for example the evidence of Liouville's theorem on conformal alterations, which isn't to be present in the traditional literature, used to be communicated to us via V. A. Zorie. The editor D. B. Fuks simplified the proofs of a number of theorems. we're thankful additionally to O. T. Bogojavlenskii, M. I. Monastyrskii, S. G. Gindikin, D. V. Alekseevskii, I. V. Gribkov, P. G. Grinevie, and E. B. Vinberg. x Preface Translator's acknowledgments. thank you are as a result of Abe Shenitzer for far type suggestion and encouragement, to numerous others of my colleagues for placing their services at my disposal, and to Eadie Henry for her first-class typing and nice persistence. Contents bankruptcy 1 Examples of Manifolds §1. the concept that of a manifold 1. 1. Definition of a manifold 1. 2. Mappings of manifolds; tensors on manifolds 1. three. Embeddings and immersions of manifolds. Manifolds with boundary §2. the best examples of manifolds 2. 1. Surfaces in Euclidean house. Transformation teams as manifolds 2. 2. Projective areas 2. three. workouts §3. crucial evidence from the idea of Lie teams three. 1. The constitution of a neighbourhood of the id of a Lie workforce. The Lie algebra of a Lie crew. Semisimplicity three. 2. the idea that of a linear illustration. An instance of a non-matrix Lie crew §4. advanced manifolds four. 1. Definitions and examples four. 2. Riemann surfaces as manifolds §5. the easiest homogeneous areas five. 1. motion of a gaggle on a manifold . five. 2. Examples of homogeneous areas five. three. workouts §6. areas of continuous curvature (symmetric areas) 6. 1. the concept that of a symmetric house 6. 2. The isometry workforce of a manifold. homes of its Lie algebra 6. three. Symmetric areas of the 1st and moment varieties 6. four. Lie teams as symmetric areas 6. five. developing symmetric areas. Examples 6. 6. workouts 1 1 I five nine 10 10 15 19 20 20 28 31 31 37 forty-one forty-one forty two forty six forty six forty six forty nine fifty one fifty three S5 S8 XII §7. Vector bundles on a manifold 7. 1. buildings related to tangent vectors 7. 2. the traditional vector package deal on a submanifold Contents fifty nine fifty nine sixty two bankruptcy 2 Foundational Questions. crucial proof relating capabilities on a Manifold. commonplace soft Mappings §8. walls of harmony and their purposes eight. 1. walls of cohesion eight. 2. the best purposes of walls of solidarity. Integrals over a manifold and the overall Stokes formulation eight. three. Invariant metrics §9. the conclusion of compact manifolds as surfaces in lR/Y §1O. numerous homes of delicate maps of manifolds 10. 1. Approximation of constant mappings by way of delicate ones 10. 2. Sard's theorem 10. three. Transversal regularity 10. four. Morse capabilities §11. purposes of Sard's theorem eleven. 1. The life of embeddings and immersions eleven.