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

By José Natário, Leonor Godinho

ISBN-10: 3319086669

ISBN-13: 9783319086668

In contrast to many different texts on differential geometry, this textbook additionally bargains fascinating functions to geometric mechanics and common relativity.

The first half is a concise and self-contained advent to the fundamentals of manifolds, differential varieties, 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 principles are illustrated and additional constructed through quite a few examples and over three hundred routines. unique ideas are supplied for lots of of those routines, making An creation to Riemannian Geometry perfect for self-study.

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

Sample text

We The isomorphism X L (G) = call g the Lie algebra of the Lie group G. 2 (1) If G = G L(n), then gl(n) = TI G L(n) = Mn×n is the space of n × n matrices with real coefficients, and the Lie bracket on gl(n) is the commutator of matrices [A, B] = AB − B A. In fact, if A, B ∈ gl(n) are two n × n matrices, the corresponding left-invariant vector fields are given by X gA = (d L g ) I (A) = x ik a k j ∂ ∂x i j x ik bk j ∂ , ∂x i j i,k, j X gB = (d L g ) I (B) = i,k, j where g ∈ G L(n) is a matrix with components x i j .

2 in Chap. 3). Show that if V1 , . . , Vn : I → M are smooth vector fields along c such that {V1 (t), . . , Vn (t)} is a basis of Tc(t) M for all t ∈ I then all these bases have the same orientation. (5) We can see the Möbius band as the 2-dimensional submanifold of R3 given by the image of the immersion g : (−1, 1) × R → R3 defined by g(t, ϕ) = 1 + t cos ϕ 2 cos ϕ, 1 + t cos ϕ 2 sin ϕ, t sin ϕ 2 . Show that the Möbius band is not orientable. (6) Let f : M → N be a diffeomorphism between two smooth manifolds.

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 .

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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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