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 thought of Riemannian Geometry a nonetheless very energetic examine sector of arithmetic. this can be a topic without loss of fascinating examples. they're certainly the major to an excellent realizing of it and may consequently play an important function all through this paintings. Of unique curiosity are the classical Lie teams permitting concrete calculations of a number of the summary notions at the menu.

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On the m-dimensional real vector space Rm we have the well-known differential operator ∂ : C ∞ (T Rm ) × C ∞ (T Rm ) → C ∞ (T Rm ) mapping a pair of vector fields X, Y on Rm to the directional derivative ∂XY of Y in the direction of X given by Y (x + tX(x)) − Y (x) . t The most fundamental properties of the operator ∂ are expressed by the following. If λ, µ ∈ R, f, g ∈ C ∞ (Rm ) and X, Y, Z ∈ C ∞ (T Rm ) then (i) ∂X(λ · Y + µ · Z) = λ · ∂XY + µ · ∂XZ, (ii) ∂X(f · Y ) = (∂Xf ) · Y + f · ∂XY , (iii) ∂(f · X + g · Y )Z = f · ∂XZ + g · ∂Y Z.

This inherits the induced metric from R(m+1)×(m+1) and the map φ : S m → RP m is what is called an isometric double cover of RP m . Long before John Nash became famous in Hollywood he proved the next remarkable result in his paper The embedding problem for Riemannian manifolds, Ann. of Math. 63 (1956), 20-63. It implies that every Riemannian manifold can be realized as a submanifold of a Euclidean space. The original proof of Nash was later simplified, see for example Matthias Gunther, On the perturbation problem associated to isometric embeddings of Riemannian manifolds, Annals of Global Analysis and Geometry 7 (1989), 69-77.

Let m be a positive integer and S m be the mdimensional unit sphere in Rm+1 . For a point p ∈ S m let Lp = {(s · p) ∈ Rm+1 | s ∈ R} be the line through the origin generated by p and ρp : Rm+1 → Rm+1 be the reflection about the line Lp . e. the set of linear endomorphisms of Rm+1 which can be identified with R(m+1)×(m+1) . It is easily checked that the reflection about the line Lp is given by ρp : q → 2 q, p p − q. It then follows from the equations ρp (q) = 2 q, p p − q = 2p p, q − q = (2ppt − e)q that the matrix in R(m+1)×(m+1) corresponding to ρp is just (2ppt − e).

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