Read e-book online 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 deals 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 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 extra built by means of a number of examples and over three hundred workouts. targeted ideas are supplied for plenty of of those workouts, 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)

Sample text

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.

Considering a parameterization ϕ : U ⊂ Rn → M on M, the integral curve c is locally given by cˆ := ϕ−1 ◦ c. Applying (dϕ−1 )c(t) to both sides of the equation defining c, we obtain ˙ˆ = Xˆ c(t) c(t) ˆ , where Xˆ = dϕ−1 ◦ X ◦ ϕ is the local representation of X with respect to the parameterizations (U, ϕ) and (T U, dϕ) on M and on T M (cf. Fig. 15). This equation is just a system of n ordinary differential equations: d cˆi (t) = Xˆ i c(t) ˆ , for i = 1, . . , n. 5 Let M be a smooth manifold and let X ∈ X(M) be a smooth vector field on M.

30 1 Differentiable Manifolds X c M ϕ Rn U cˆ ˆ X Fig. 2), depends smoothly on the initial point p (see [Arn92]). 6 Let X ∈ X(M). For each p ∈ M there exists a neighborhood W of p, an interval I = (−ε, ε) and a mapping F : W × I → M such that: (i) for a fixed q ∈ W the curve F(q, t), t ∈ I , is an integral curve of X at q, that is, F(q, 0) = q and ∂∂tF (q, t) = X F(q,t) ; (ii) the map F is differentiable. The map F : W × I → M defined above is called the local flow of X at p. Let us now fix t ∈ I and consider the map ψt : W → M q → F(q, t) = cq (t).

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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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