By Michael Spivak

Booklet by way of Michael Spivak, Spivak, Michael

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**A Comprehensive Introduction to Differential Geometry**

Ebook by way of Michael Spivak, Spivak, Michael

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Then ∇X Y = (XY k + X i Y j Γkij )Ek . Proof. Just use the deﬁning rules for a connection and compute: ∇X Y = ∇X (Y j Ej ) = (XY j )Ej + Y j ∇X i Ei Ej = (XY j )Ej + X i Y j ∇Ei Ej = XY j Ej + X i Y j Γkij Ek . 3). 3) 52 4. Connections Existence of Connections So far, we have studied properties of connections, but have not produced any, so you might be wondering if they are plentiful or rare. In fact, they are quite plentiful, as we will show shortly. Let’s begin with a trivial example: on Rn , deﬁne the Euclidean connection by ∇X Y j ∂j = (XY j )∂j .

2). More generally, if V is any n-dimensional vector space endowed with an inner product, we can set g(X, Y ) = X, Y for any X, Y ∈ Tp V = V . Choosing an orthonormal basis (E1 , . . , En ) for V deﬁnes a map from Rn to V by sending (x1 , . . , xn ) to xi Ei ; this is easily seen to be an isometry of (V, g) with (Rn , g¯). Spheres Our second model space is the sphere of radius R in Rn+1 , denoted SnR , ◦ with the metric g R induced from the Euclidean metric on Rn+1 , which we call the round metric of radius R.

We can immediately write down a large group of isometries of SnR by observing that the linear action of the orthogonal group O(n + 1) on Rn+1 preserves SnR and the Euclidean metric, so its restriction to SnR acts by isometries of the sphere. 3. O(n + 1) acts transitively on orthonormal bases on SnR . More precisely, given any two points p, p˜ ∈ SnR , and orthonormal bases {Ei } 34 3. 2. Transitivity of O(n + 1) on orthonormal bases. ˜i } for Tp˜Sn , there exists ϕ ∈ O(n + 1) such that ϕ(p) = p˜ for Tp SnR and {E R ˜ and ϕ∗ Ei = Ei .