# An Introduction to Riemannian Geometry by Gudmundsson S.

By Gudmundsson S.

Those lecture notes grew out of an M.Sc. direction on differential geometry which I gave on the collage of Leeds 1992. Their major function is to introduce the attractive conception of Riemannian Geometry a nonetheless very lively learn sector of arithmetic. it is a topic with out loss of attention-grabbing examples. they're certainly the main to an outstanding knowing of it and should for this reason play a massive position all through this paintings. Of certain curiosity are the classical Lie teams permitting concrete calculations of the various summary notions at the menu.

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X1 ∂xm Let X, Y ∈ C ∞ (T Rm ) be two vector fields given by m ∂ X= αk and Y = ∂xk k=1 ∞ m m βk k=1 ∂ , ∂xk where αk , βk ∈ C (R ). Find a formula for the Lie bracket [X, Y ] in terms of the standard global frame. CHAPTER 5 Riemannian Manifolds In this chapter we introduce the important notion of a Riemannian metric on a differentiable manifold. This is the most important example of what is called a tensor field. The metric provides us with an inner product on each tangent space and can be used to measure the length of curves in the manifold.

6. We shall now determine the normal bundle N O(m) of the orthogonal group O(m) as a submanifold of Rm×m . 20. The orthogonal group O(m) is a subset of the linear space Rm×m equipped with the Riemannian metric X, Y = trace(X t · Y ) inducing a left invariant metric on O(m). We have already seen that the tangent space Te O(m) of O(m) at the neutral element e is Te O(m) = {X ∈ Rm×m |X + X t = 0} and that the tangent bundle T O(m) of O(m) is given by T O(m) = {(x, xX)| x ∈ O(m), X ∈ Te O(m)}. The space Rm×m of real m × m matrices has a linear decomposition Rm×m = Sym(Rm ) ⊕ Te O(m) and every element X ∈ Rm×m can be decomposed X = X + X ⊥ in its symmetric and skew-symmetric parts given by X = (X − X t )/2 and X ⊥ = (X + X t )/2.

15. CHAPTER 7 Geodesics In this chapter we introduce the notion of a geodesic on a smooth manifold as a solution to a non-linear system of ordinary differential equations. We then show that geodesics are solutions to two different variational problems. They are critical points to the so called energy functional and furthermore locally shortest paths between their endpoints. 1. Let M be a smooth manifold and (T M, M, π) be its tangent bundle. A vector field X along a curve γ : I → M is a curve X : I → T M such that π ◦ X = γ.