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

By José Natário, Leonor Godinho

Not like many different texts on differential geometry, this textbook additionally bargains attention-grabbing purposes to geometric mechanics and normal relativity.

The first half is a concise and self-contained advent 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 principles are illustrated and additional built by means of a number of examples and over three hundred routines. designated ideas are supplied for plenty of of those workouts, making An creation to Riemannian Geometry excellent for self-study.

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N). Proof Let us consider the coordinate chart x = x 1 , . . , x n . 6 Vector Fields 27 ∂ ∂x 1 X p = X 1 ( p) ∂ ∂x n + · · · + X n ( p) p p for some functions X i : W → R. In the local chart associated with the parameterization (U × Rn , dϕ) of T M, the local representation of the map X is Xˆ x 1 , . . , 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.

Again, the derivative of f is surjective at a point A ∈ G L(n), making S L(n) into a Lie group. Indeed, it is easy to see that det (I + h B) − det I = tr B h→0 h (d f ) I (B) = lim implying that det (A + h B) − det A h→0 h (det A) det I + h A−1 B − det A = lim h→0 h det I + h A−1 B − 1 = (det A) lim h→0 h −1 = (det A) (d f ) I (A B) = (det A) tr(A−1 B). (d f ) A (B) = lim Since det (A) = 1, for any k ∈ R, we can take the matrix B = nk A to obtain (d f ) A (B) = tr nk I = k. Therefore, (d f ) A is surjective for every A ∈ S L(n), and so 1 is a regular value of f .

X ∂y (c) Given V, W ∈ h, compute [V, W ]. (d) Determine the flow of the vector field X V , and give an expression for the exponential map exp : h → H . (e) Confirm your results by first showing that H is the subgroup of G L(2) formed by the matrices yx 01 with y > 0. 7 Lie Groups 45 which we already know to be a 3-manifold. Making a = p + q, d = p − q, b = r + s, c = r − s, show that S L(2) is diffeomorphic to S 1 × R2 . (5) Give examples of matrices A, B ∈ gl(2) such that e A+B = e A e B . (6) For A ∈ gl(n), consider the differentiable map h : R → R\{0} t → det e At and show that: (a) this map is a group homomorphism between (R, +) and (R\{0}, ·); (b) h (0) = trA; (c) det(e A ) = etrA .

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