By José Natário, Leonor Godinho

ISBN-10: 3319086669

ISBN-13: 9783319086668

Not like many different texts on differential geometry, this textbook additionally deals fascinating purposes to geometric mechanics and common 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 reports 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 extra constructed through a number of examples and over three hundred routines. certain options are supplied for lots of of those routines, making An creation to Riemannian Geometry perfect for self-study.

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

Example text

Then H acts on G by left multiplication: A(h, g) = h · g for h ∈ H , g ∈ G. (2) G L(n) acts on Rn through A · x = Ax for A ∈ G L(n) and x ∈ Rn . The same is true for any subgroup G ⊂ G L(n). For each p ∈ M we can define the orbit of p as the set G · p := {g · p | g ∈ G}. If G · p = { p} then p is called a fixed point of G. e. G · p = M), then the action is said to be transitive. Note that when this happens, there is only one orbit and, for every p, q ∈ M with p = q, there is always an element of the group g ∈ G such that q = g · p.

X m , x n+1 , . . , x m . Its derivative at 0 is the linear map given by ⎛ ⎞ In×n | ∗ ⎠. (d F)0 = ⎝ − − − + − − − 0 | I(m−n)×(m−n) The inverse function theorem then implies that F is a local diffeomorphism at 0, meaning that it maps some open neighborhood of this point U ⊂ U , diffeomorphically onto an open set W of Rm containing 0. If π1 : Rm → Rn is the standard projection onto the first n factors, we have π1 ◦ F = fˆ, and hence fˆ ◦ F −1 = π1 : W → Rn . Therefore, replacing ϕ by ϕ := ϕ ◦ F −1 , we obtain coordinates for which f is the standard projection π1 onto the first n factors: ψ −1 ◦ f ◦ ϕ = ψ −1 ◦ f ◦ ϕ ◦ F −1 = fˆ ◦ F −1 = π1 .

4) The map f : G L(n) → R given by f (A) = det A is differentiable, and the level set f −1 (1) is S L(n) = {A ∈ Mn×n | det A = 1}, the special linear group. 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).

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An Introduction to Riemannian Geometry: With Applications to Mechanics and Relativity (Universitext) by José Natário, Leonor Godinho


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