By José Natário, Leonor Godinho

ISBN-10: 3319086669

ISBN-13: 9783319086668

Not like 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 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 built by way of a number of examples and over three hundred routines. unique options are supplied for plenty of of those workouts, making An creation to Riemannian Geometry perfect for self-study.

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

Sample text

5) Let φ : (0, +∞) × (0, π) × (0, 2π) → R3 be the parameterization of U = R3 \ {(x, 0, z) | x ≥ 0 and z ∈ R} by spherical coordinates, φ(r, θ, ϕ) = (r sin θ cos ϕ, r sin θ sin ϕ, r cos θ). ∂ ∂ ∂ , ∂θ and ∂ϕ at each point of U . Determine the Cartesian components of ∂r (6) Compute the derivative (d f ) N of the antipodal map f : S n → S n at the north pole N . (7) Let W be a coordinate neighborhood on M, let x : W → Rn be a coordinate chart and consider a smooth function f : M → R. Show that for p ∈ W , the derivative (d f ) p is given by (d f ) p = ∂ fˆ (x( p)) d x 1 ∂x 1 p + ··· + ∂ fˆ (x( p)) d x n ∂x n p , where fˆ := f ◦ x −1 .

If this curve has initial value c(0) = p, we denote it by c p and we say that c p is an integral curve of X at p. 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, .

1 Let W be a coordinate neighborhood on M (that is, W = ϕ(U ) for some parameterization ϕ : U → M), and let x := ϕ−1 : W → Rn be the corresponding coordinate chart. Then a map X : W → T W is a differentiable vector field on W if and only if, X p = X 1 ( p) ∂ ∂x 1 + · · · + X n ( p) p ∂ ∂x n p for some differentiable functions X i : W → R (i = 1, . . , n). Proof Let us consider the coordinate chart x = x 1 , . . , x n . 6 Vector Fields 27 ∂ ∂x 1 X p = X 1 ( p) ∂ ∂x n + · · · + X n ( p) p p for some functions X i : W → R.

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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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