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An Introduction to Riemannian Geometry: With Applications to by José Natário, Leonor Godinho

By José Natário, Leonor Godinho

In contrast to many different texts on differential geometry, this textbook additionally deals 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 reports 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 extra built via a variety of examples and over three hundred routines. particular ideas are supplied for lots of of those routines, making An advent to Riemannian Geometry perfect for self-study.

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Additional info for An Introduction to Riemannian Geometry: With Applications to Mechanics and Relativity (Universitext)

Sample text

However, the commutator X ◦ Y − Y ◦ X does define a vector field. 2 Given two differentiable vector fields X, Y ∈ X(M) on a smooth manifold M, there exists a unique differentiable vector field Z ∈ X(M) such that Z · f = (X ◦ Y − Y ◦ X ) · f for every differentiable function f ∈ C ∞ (M). Proof Considering a coordinate chart x : W ⊂ M → Rn , we have n X= Xi i=1 ∂ ∂x i n and Y = Yi i=1 ∂ . ∂x i Then, n (X ◦ Y − Y ◦ X ) · f =X· Yi i=1 ∂ fˆ ∂x i n X · Yi = i=1 n −Y · Xi i=1 ∂ fˆ ∂x i ˆ ∂ fˆ i ∂f − Y · X ∂x i ∂x i 28 1 Differentiable Manifolds n + X jYi i, j=1 2 ˆ ∂ 2 fˆ j i ∂ f − Y X ∂x j ∂x i ∂x j ∂x i n = X · Y i − Y · Xi i=1 ∂ ∂x i · f, and so, at each point p ∈ W , one has ((X ◦ Y − Y ◦ X ) · f ) ( p) = Z p · f , where n X · Y i − Y · X i ( p) Zp = i=1 ∂ ∂x i .

9) Let f : M → N be a diffeomorphism between smooth manifolds. Show that f ∗ [X, Y ] = [ f ∗ X, f ∗ Y ] for every X, Y ∈ X(M). Therefore, f ∗ induces a Lie algebra isomorphism between X(M) and X(N ). (10) Let f : M → N be a differentiable map between smooth manifolds and consider two vector fields X ∈ X(M) and Y ∈ X(N ). Show that: (a) if the vector field Y is f -related to X then any integral curve of X is mapped by f into an integral curve of Y ; (b) the vector field Y is f -related to X if and only if the local flows FX and FY satisfy f (FX ( p, t)) = FY ( f ( p), t) for all (t, p) for which both sides are defined.

Bn }. There is a unique linear transformation S : V → V such that bi = S bi for every i = 1, . . , n. We say that the two bases are equivalent if det S > 0. This defines an equivalence relation that divides the set of all ordered bases of V into two equivalence classes. An orientation for V is an assignment of a positive sign to the elements of one equivalence class and a negative sign to the elements of the other. The sign assigned to a basis is called its orientation and the basis is said to be positively oriented or negatively oriented according to its sign.

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