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An Introduction to Riemannian Geometry by Gudmundsson S.

By Gudmundsson S.

Those lecture notes grew out of an M.Sc. path on differential geometry which I gave on the college of Leeds 1992. Their major goal is to introduce the gorgeous conception of Riemannian Geometry a nonetheless very lively examine region of arithmetic. it is a topic without loss of fascinating examples. they're certainly the most important to an outstanding realizing of it and may accordingly play an incredible position all through this paintings. Of distinctive curiosity are the classical Lie teams permitting concrete calculations of some of the summary notions at the menu.

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Let m be a positive integer. Use the Hairy Ball Theorem to prove that the tangent bundles T S 2m of the even-dimensional spheres S 2m are not trivial. Construct a non-vanishing vector field X ∈ C ∞ (T S 2m+1 ) on the odd-dimensional sphere S 2m+1 . 4. 13. 5. , }. ∂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.

6 to prove that the Hopf map φ : S 3 → S 2 with φ : (x, y) → (2x¯ y , |x|2 − |y|2 ) is a submersion. CHAPTER 4 The Tangent Bundle The main aim of this chapter is to introduce the tangent bundle T M of a differentiable manifold M m . Intuitively this is the object we get by glueing at each point p ∈ M the corresponding tangent space Tp M . The differentiable structure on M induces a differentiable structure on T M making it into a differentiable manifold of dimension 2m. The tangent bundle T M is the most important example of what is called a vector bundle over M .

Prove that the second fundamental form B of M in N is symmetric and tensorial in both its arguments. 7. 14. 8. 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.

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