Lectures on Kähler Geometry (London Mathematical Society Student Texts)

This definition has a pleasant geometrical which means, yet in perform one frequently makes use of the next replacement characterization: Proposition 6. nine. enable ξ be a vector box on a Riemannian manifold (M, g, ∇). the subsequent statements are identical: (i) ξ is Killing with appreciate to g. (ii) The Lie spinoff of g with appreciate to ξ vanishes: Lξ g = zero. (iii) The covariant spinoff ∇ξ is skew-symmetric with recognize to g: g(∇X ξ, Y ) + g(∇Y ξ, X) = zero, ∀ X, Y ∈ T M. (6. four) facts. allow ϕt denote the neighborhood circulate of ξ. If ξ is Killing, ϕt are neighborhood isometries so via (2. 6) Lξ g = − d dt t=0 (ϕt )∗ (g) = d dt t=0 (ϕt )∗ (g) = d dt g = zero. t=0 Conversely, feel that Lξ g = zero. seeing that (ϕt )∗ ξ = ξ, the naturality of the Lie spinoff (2. 7) yields zero = (ϕs )∗ (Lξ g) = Lξ ((ϕs )∗ g) 52 6. Riemannian manifolds for all fastened s. taking into consideration the truth that ϕt is a pseudogroup of neighborhood diffeomorphisms (1. 7) we get zero = Lξ ((ϕs )∗ g) = d dt t=0 (ϕt )∗ ((ϕs )∗ g) = d dt t=0 (ϕs+t )∗ (g) d (ϕt )∗ g. dt t=s This exhibits that the relations of tensors (ϕt )∗ g doesn't rely on t, so (ϕt )∗ g = (ϕ0 )∗ g = g, hence proving the equivalence of (i) and (ii). = The equivalence of (ii) and (iii) is a final result of Lemma 6. four. comment. The isometry staff of each compact manifold is a Lie staff, whose Lie algebra is isomorphic to the gap of Killing vector fields. 6. five. workouts (1) convey that the holonomy team Hol(u) at a few orthonormal body u of a Riemannian manifold (M n , g) is inside the orthogonal workforce On . (2) (Conformal metric alterations) allow (M, g) be a Riemannian manifold and allow u : M → R be any functionality. We outline a brand new Riemannian metric on M through g = e2u g. express that the Levi–Civita covariant derivatives ∇ and ∇ of g and g are comparable by way of ∇X Y = ∇X Y + (∂X u)Y + (∂Y u)X − g(X, Y )Gradg u , the place Gradg u is the twin of du with appreciate to g. (3) allow (M, g, ∇) be a Riemannian manifold. If η denotes a 1-form on M , we may possibly view its covariant by-product ∇η as a bilinear shape on T M by means of the formulation ∇η(X, Y ) := (∇X η)(Y ). exhibit that the outside by-product of η equals two times the skew-symmetric a part of ∇η: dη(X, Y ) = ∇η(X, Y ) − ∇η(Y, X). (4) turn out that the set of Killing vector fields is a Lie algebra with appreciate to the standard Lie bracket. trace: Use (2. nine) and Proposition 6. nine. (5) (Kostant formulation) allow ξ be a Killing vector box on a Riemannian manifold (M, g, ∇) with Riemannian curvature tensor R. turn out that the covariant by-product of the (1,1)-tensor ∇ξ satisfies ∇X (∇ξ) = R(X, ξ), ∀ X ∈ T M. 6. five. workouts fifty three trace: If X, Y, Z are vector fields on M , take the covariant spinoff of (6. four) with recognize to Z and use round variations in X, Y, Z. (6) exhibit Killing vector box ξ is dependent upon its 1-jet at any element. trace: Use the former workout to teach that (ξ, ∇ξ) is a parallel component of the package deal T M ⊕ End(T M ) with appreciate to a undeniable covariant spinoff on that package, and practice Lemma five. four to finish that if ξ and ∇ξ either vanish sooner or later, then ξ is identically 0.