By José Natário, Leonor Godinho
Not like many different texts on differential geometry, this textbook additionally deals fascinating purposes to geometric mechanics and normal 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 purposes 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 built by means of a variety of examples and over three hundred workouts. designated strategies are supplied for lots of of those workouts, making An advent to Riemannian Geometry excellent for self-study.
Read Online or Download An Introduction to Riemannian Geometry: With Applications to Mechanics and Relativity (Universitext) PDF
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Additional resources for An Introduction to Riemannian Geometry: With Applications to Mechanics and Relativity (Universitext)
X n = x 1 , . . , x n , Xˆ 1 x 1 , . . , x n , . . , Xˆ n x 1 , . . , x n . e. if and only if the functions X i : W → R are differentiable. A vector field X is differentiable if and only if, given any differentiable function f : M → R, the function X· f :M →R p → X p · f := X p ( f ) is also differentiable [cf. 11(1)]. This function X · f is called the directional derivative of f along X . Thus one can view X ∈ X(M) as a linear operator X : C ∞ (M) → C ∞ (M). Let us now take two vector fields X, Y ∈ X(M).
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 .
When dim M < dim N , the best we can hope for is that (d f ) p : T p M → T f ( p) N is injective. The map f is then called an immersion at p. If f is an immersion at every point in M, it is called an immersion. Locally, every immersion is (up to a diffeomorphism) the canonical immersion of Rm into Rn (m < n) where a point x 1 , . . , x m is mapped to x 1 , . . , x m , 0, . . , 0 . This result is known as the local immersion theorem. 1 Let f : M → N be an immersion at p ∈ M. Then there exist local coordinates around p and f ( p) on which f is the canonical immersion.