Algebraic Geometry - Bowdoin 1985, Part 2 by Bloch S. (ed.)

By Bloch S. (ed.)

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Proof. (i) Clearly e ∈ Hol(u). If a, b ∈ Hol(u), let γ and γ be two horizontal paths in P joining u with ua and ub respectively. By right invariance, the composition γ · (γb) is a piecewise smooth horizontal path joining u with uab. Similarly, γ −1 a−1 joins u with ua−1 , where γ −1 (t) denotes the path γ(1 − t). (ii) For every b ∈ Hol(u) there exists a horizontal path γ joining u with ub. The horizontal path γa then joins ua with uba = ua(a−1 ba), thus showing that a−1 Hol(u)a ⊂ Hol(ua). The other inclusion is similar.

Ii) The Lie derivative of g with respect to ξ vanishes: Lξ g = 0. (iii) The covariant derivative ∇ξ is skew-symmetric with respect to g: g(∇X ξ, Y ) + g(∇Y ξ, X) = 0, ∀ X, Y ∈ T M. 4) Proof. Let ϕt denote the local flow of ξ. 6) Lξ g = − d dt t=0 (ϕt )∗ (g) = d dt t=0 (ϕt )∗ (g) = d dt g = 0. t=0 Conversely, suppose that Lξ g = 0. 7) yields 0 = (ϕs )∗ (Lξ g) = Lξ ((ϕs )∗ g) 52 6. Riemannian manifolds for all fixed s. 7) we get 0 = Lξ ((ϕs )∗ g) = d dt t=0 (ϕt )∗ ((ϕs )∗ g) = d dt t=0 (ϕs+t )∗ (g) d (ϕt )∗ g.

3. Around every point x in a Riemannian manifold (M n , g) there exists a local orthonormal frame u = {e1 , . . , en } parallel at x with respect to ∇. Let (M n , g) be an oriented Riemannian manifold. Since the representation of Gln (R) on Λn Rn is given by the determinant, its restriction to SOn is trivial, thus showing that the vector bundle Λn M has a distinguished section dv := [u, ζ] where u is any oriented orthonormal frame on M and ζ ∈ Λn Rn is the canonical unit element. The n-form dv is called the volume form.

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