HOMEWORK #4 - DUE FEB 21, AT NOON

Exercise 1 (Torsion tensor). Prove that the torsion tensor T (X,Y ) = ∇X Y − ∇Y X − [X,Y ] is a (2,1)-tensor (Hint: use ex. 1 from last week). Exercise 2 (Difference tensor). Given two connections ∇0, ∇1 on TM, show that: (1) The sum ∇0 + ∇1 is not a connection, while the sum a∇0 + b∇1 is a connection, provided that a + b = 1. 1 0 (2) The map A(X,Y ) = ∇X Y − ∇X Y defines a (2, 1)-tensor field (called dif- ference tensor). (3) The set of all connections on TM is precisely ∇0 + Γ(T 2,1M). (4) The connections have the same geodesics if and only if the difference tensor is skew symmetric (A(X,Y ) = −A(Y,X)). (5) The connections have the same torsion if an only if the difference tensor is symmetric (A(X,Y ) = A(Y,X)).

Exercise 3 (Naturality of Levi Civita connection). Let (M, h, iM ), (N, h, iN ) be Riemannian manifolds, with Levi Civita connections ∇M , ∇N respectively. Sup- pose there is a diffeomorphism φ : M → N. (1) Check that for every vector field X ∈ X (M), one can define a push- forward vector field φ∗X ∈ X (N) by (φ∗X)p := φ∗(Xφ−1(p)). Prove that φ∗[X,Y ] = [φ∗X, φ∗Y ]. (2) Prove that if φ is an isometry, then φ (∇M Y ) := ∇N φ Y for every ∗ X φ∗X ∗ X,Y ∈ X (M). Exercise 4 (A natural metric on the tangent bundle). Let (M, h, i) be a Rie- mannian manifold, and π : TM → M the tangent bundle. Recall that a curve α : (0, 1) → TM has the form α(t) = (p(t), v(t)). In other words, you can think of α as a curve p(t) = π(α(t)) on M, together with a vector field v(t) along p(t). Given vectors V,W ∈ T(p,v)(TM), take curves α(t) = (p(t), v(t)), β(t) = (q(t), w(t)) with α(0) = β(0) = (p, v) and V = α0(0), W = β0(0), and define  D D 

V,W (p,v) := hπ∗V, π∗W ip + v(t), w(t) . dt t=0 dt t=0 p (1) Prove that the definition does not depend on the choice of curves α, β, and it defines a Riemannian metric on TM. −1 (2) Given (p, v) ∈ TM, recall that the fiber Fp := π (p) passing through p, it simply the the tangent space TpM (which we think of as a sub- manifold of TM). Define the horizontal space at (p, v) as the subspace H(p,v) ⊂ T(p,v)(TM) perpendicular to T(p,v)Fp in the metric above. Prove that d(p,v)π defines a linear isomorphism from H(p,v) to TpM. 0 (3) A curve α(t) = (p(t), v(t)) is called horizontal if for every t, α (t) ∈ Hα(t). Prove that α is horizontal iff v(t) is parallel along p(t). Exercise 5 (Isometries). Given an isometry φ : M → N between Riemannian manifolds, prove that φ takes geodesics to geodesics. More precisely, φ(expp tv) = expφ(p) tφ∗v. 1