José Natário, Leonor Godinho's An Introduction to Riemannian Geometry: With Applications to PDF

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 basic 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 reviews 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 extra constructed through various examples and over three hundred workouts. certain recommendations are supplied for plenty of of those routines, making An advent to Riemannian Geometry excellent for self-study.

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

Example text

We can obtain differentiable manifolds by taking inverse images of regular values. 6 Let q ∈ N be a regular value of f : M → N and assume that the level set L := f −1 (q) = { p ∈ M | f ( p) = q} is nonempty. Then L is a submanifold of M and T p L = ker(d f ) p ⊂ T p M for all p ∈ L. Proof For each point p ∈ f −1 (q), we choose parameterizations (U, ϕ) and (V, ψ) around p and q for which f is the standard projection π1 onto the first n factors, ϕ(0) = p and ψ(0) = q (cf. 4). 5 Immersions and Embeddings 25 ϕ−1 f −1 (q) = π1−1 ψ −1 (q) = π1−1 (0) = and so U := ϕ−1 (L) = 0, .

44 1 Differentiable Manifolds (2) (a) Show that (Rn , +) is a Lie group, determine its Lie algebra and write an expression for the exponential map. (b) Prove that, if G is an abelian Lie group, then [V, W ] = 0 for all V, W ∈ g. (3) We can identify each point in H = (x, y) ∈ R2 | y > 0 with the invertible affine map h : R → R given by h(t) = yt + x. The set of all such maps is a group under composition; consequently, our identification induces a group structure on H . (a) Show that the induced group operation is given by (x, y) · (z, w) = (yz + x, yw), and that H , with this group operation, is a Lie group.

When the interval of definition I of cq is R, this local 1-parameter group of diffeomorphisms becomes a group of diffeomorphisms. A vector field X whose local flow defines a 1-parameter group of diffeomorphisms is said to be complete. This happens for instance when the vector field X has compact support. 8 If X ∈ X(M) is a smooth vector field with compact support then it is complete. Proof For each p ∈ M we can take a neighborhood W and an interval I = (−ε, ε) such that the local flow of X at p, F(q, t) = cq (t), is defined on W × I .

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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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