By José Natário, Leonor Godinho

In contrast to 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 creation to the fundamentals of manifolds, differential kinds, metrics and curvature. the second one half stories 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 principles are illustrated and additional built through various examples and over three hundred workouts. specified options are supplied for plenty of of those workouts, making An creation 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)

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K! In fact, this series converges for any matrix A and the map h(t) = e At satisfies h(0) = e0 = I dh (t) = e At A = h(t)A. dt Hence, h is the flow of X A at the identity (that is, h(t) = ψt (e)), and so exp A = ψ1 (e) = e A . 40 1 Differentiable Manifolds Let now G be any group and M be any set. We say that G acts on M if there is a homomorphism φ from G to the group of bijective mappings from M to M, or, equivalently, writing φ(g)( p) = A(g, p), if there is a mapping A : G × M → M satisfying the following conditions: (i) if e is the identity in G, then A(e, p) = p, ∀ p ∈ M; (ii) if g, h ∈ G, then A(g, A(h, p)) = A(gh, p), ∀ p ∈ M.

If M ⊂ N and the inclusion map i : M → N is an embedding, M is said to be a submanifold of N . Therefore, an embedding f : M → N maps M diffeomorphically onto a submanifold of N . Charts on f (M) are just restrictions of appropriately chosen charts on N to f (M) [cf. 9(3)]. A differentiable map f : M → N for which (d f ) p is surjective is called a submersion at p. Note that, in this case, we necessarily have m ≥ n. If f is a submersion at every point in M it is called a submersion. Locally, every submersion is the standard projection of Rm onto the first n factors.

X n−1 , 0 = = y 1 x 1 , . . , x n−1 , 0 , . . , y n−1 x 1 , . . , x n−1 , 0 , 0 and ϕ−1 β ◦ ϕα x 1 , . . , x n−1 = y 1 x 1 , . . , x n−1 , 0 , . . , y n−1 x 1 , . . , x n−1 , 0 . Consequently, denoting x 1 , . . , x n−1 , 0 by (x, 0), ⎛ d ϕ−1 β ◦ ϕα (x,0) ⎜ =⎝ and so det d ϕ−1 β ◦ ϕα (x,0) = d ϕ−1 β ◦ ϕα −−− 0 x | ∗ ⎞ ⎟ + −−− ⎠ n | ∂∂xy n (x, 0) ∂ yn (x, 0) det d ϕ−1 β ◦ ϕα ∂x n x . However, fixing x 1 , · · · , x n−1 , we have that y n is positive for positive values of x n n and is zero for x n = 0.

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