By Leonor Godinho, José Natário
Unlike many different texts on differential geometry, this textbook additionally bargains attention-grabbing functions to geometric mechanics and common relativity.
The first half is a concise and self-contained creation to the fundamentals of manifolds, differential types, metrics and curvature. the second one half experiences functions to mechanics and relativity together with the proofs of the Hawking and Penrose singularity theorems. it may be independently used for one-semester classes in both of those subjects.
The major rules are illustrated and additional constructed through a variety of examples and over three hundred routines. particular ideas are supplied for lots of of those workouts, making An advent to Riemannian Geometry perfect for self-study.
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Extra resources for An Introduction to Riemannian Geometry: With Applications to Mechanics and Relativity
Locally, every immersion is (up to a diffeomorphism) the canonical immersion of into () where a point is mapped to . This result is known as the local immersion theorem . 1 Let be an immersion at . Then there exist local coordinates around and on which is the canonical immersion. Proof Let and be parameterizations around and . Let us assume for simplicity that and . Since is an immersion, is injective (where is the expression of in local coordinates). Hence we can assume (changing basis on if necessary) that this linear transformation is represented by the matrix where is the identity matrix.
3)We could also have defined -manifolds by requiring the coordinate changes to be -maps (a -manifold would then denote a topological manifold). 3 (1)The space with the usual topology defined by the Euclidean metric is a Hausdorff space and has a countable basis of open sets. 1 are trivially satisfied and we have an atlas for . The maximal atlas that contains this parameterization is usually called the standard differentiable structure on . We can of course consider other atlases. Take, for instance, the atlas defined by the parameterization with for a nonsingular -matrix .
In addition, we give a brief overview of Lie groups and Lie group actions. Chapter 2 is devoted to differential forms, covering the standard topics: wedge product, pull-back, exterior derivative, integration and the Stokes theorem. Riemannian manifolds are introduced in Chap. 3 , where we treat the Levi–Civita connection, minimizing properties of geodesics and the Hopf–Rinow theorem. Chapter 4 addresses the notion of curvature. In particular, we use the powerful computational method given by the Cartan structure equations to prove the Gauss–Bonnet theorem.
An Introduction to Riemannian Geometry: With Applications to Mechanics and Relativity by Leonor Godinho, José Natário