A Course in Differential GeometryAn introduction to differential geometry with principal emphasis on Riemannian geometry. Ch. I explains basic definitions and gives the proofs of the important theorems of Whitney and Sard. Ch. II deals with vector fields and differential forms. Ch. III addresses integration of vector fields and p-plane fields. Ch. IV develops the notion of connection on a Riemannian manifold considered as a means to define parallel transport on the manifold.The author also discusses related notions of torsion and curvature, and gives a working knowledge of the covariant derivative. Ch. V specializes on Riemannian manifolds by deducing global properties from local properties of curvature, the final goal being to determine the manifold completely. Ch. VI explores some problems in PDEs suggested by the geometry of manifolds. The text contains many problems and solutions, permitting the reader to apply the theorems and to see concrete developments of the abstract theory. |
Contents
Differentiable Manifolds | 19 |
Tangent Space | 43 |
An Introduction to Research | 47 |
Integration of Vector Fields and Differential Forms | 77 |
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Common terms and phrases
according atlas Banach space basis bijection changes of charts choose Christoffel symbols Ck-differentiable compact set components Compute connected consider the map coordinate system covariant curvature Deduce defined Definition denote diffeomorphism differentiable curve differentiable function differentiable manifold differentiable map differential forms dt dt dx¹ dx² dx³ equivalence exists expp exterior differential Əxi finite geodesic homeomorphism imbedding immersion implies integral manifold Jacobian Lemma Let f Let Mn linear manifold Mn manifold of dimension matrix measure zero neighbourhood open set orientable p-forms partition of unity Problem proof Proposition prove rank real numbers Riemannian manifold Rn+1 satisfies sequence Show Solution to Exercise submanifold subset suppose tangent vector tensor theorem topological space topology Tp(M vanish vector field vector space verify მ მ მე