285G, Lecture 0: Riemannian manifolds and curvature – Exercises

Exercise 1. Show that the map (X, Y) \mapsto [X,Y] endows the space \Gamma(TM) of vector fields with the structure of an abstract Lie algebra. Also establish the Leibniz rule

{}[X, fY] = (\nabla_X f) Y + f [X,Y] (5)

for all X, Y \in \Gamma(TM) and f \in C^\infty(M). \diamond

Exercise 2. Let \nabla, \nabla' be two connections on TM. Show that there exists a unique rank (1,2) tensor \Gamma^\alpha_{\beta \gamma} = \nabla' - \nabla such that

\nabla'_\beta f^\alpha - \nabla_\beta f^\alpha = \Gamma^\alpha_{\beta \gamma} f^\gamma (13)

for all vector fields f^\alpha. Now interpret the Christoffel symbol \Gamma^a_{bc} of a connection \nabla on TM relative to a frame e = (e_a)_{a \in A} as the difference \nabla - \nabla^{(e)} of that connection with the flat connection \nabla^{(e)} induced by the trivialisation of the tangent bundle induced by that frame. \diamond

Exercise 3. Show that \nabla is torsion-free if and only if

{}[X,Y]^\alpha = X^\beta \nabla_\beta Y^\alpha - Y^\beta \nabla_\beta X^\alpha (15)

for all vector fields X, Y (or in coordinate-free notation, {}[X,Y] = \nabla_X Y - \nabla_Y X). \diamond

Exercise 4. If \nabla is a torsion-free connection on TM, and R_{\alpha \beta \gamma}^\delta is the tensor form of the curvature R, defined by requiring that

(R(X,Y) Z)^\delta = R_{\alpha \beta \gamma}^\delta X^\alpha Y^\beta Z^\gamma, (17)

then show that

\nabla_\alpha \nabla_\beta X^\delta - \nabla_\beta \nabla_\alpha X^\delta = R_{\alpha \beta \gamma}^\delta X^\gamma (18)

for all vector fields X^\delta. What is the analogue of (18) if X^\delta is replaced by a rank (k,l) tensor? \diamond

Exercise 5. Prove this theorem. (Hint: one can either (a) use abstract index notation and study expressions such as \nabla_\alpha X^\beta, (b) use coordinate-free notation and study expressions such as g( \nabla_X Y, Z ), or (c) use local coordinates (e.g. use a frame e_a := \phi^* \frac{d}{dx^a} arising from a chart \phi as in Example 2) and work with the Christoffel symbols \Gamma^a_{bc}. It is instructive to do this exercise in all three possible ways in order to appreciate the equivalence (and relative advantages and disadvantages) between these three perspectives. \diamond

Exercise 6. Show that the above three symmetries of \hbox{Riem} imply that \hbox{Riem} is a self-adjoint section of \hbox{Hom}( \bigwedge^2 TM, \bigwedge^2 TM ), and that these conditions are in fact equivalent in three and fewer dimensions. (The claim fails in four and higher dimensions; see comments.) \diamond

Exercise 7. By differentiating (24) and cyclically summing, establish the second Bianchi identity

Exercise 8. Show that a Riemannian manifold (M,g) is locally isomorphic (as Riemannian manifolds) to Euclidean space if and only if the Riemann curvature tensor vanishes. (Hint: one direction is easy. For the other direction, the quickest way is to apply the Frobenius theorem to obtain a local trivialisation of the tangent bundle which is flat with respect to the Levi-Civita connection.) This illustrates the point that the Riemann curvature captures all the local obstructions that prevent a Riemannian manifold from being flat. (Compare this situation with the superficially similar subject of symplectic geometry, in which Darboux’s theorem guarantees that there are no local obstructions whatsoever to a symplectic manifold (M,\omega) being flat.) \diamond

Exercise 9. (Ricci controls Riemann in three dimensions) In three dimensions, suppose that the (necessarily real) eigenvalues of the Riemann curvature at a point x (viewed as an element of \hbox{Hom}(\bigwedge^2 TM, \bigwedge^2 TM)) are \lambda, \mu, \nu. Show that the eigenvalues of the Ricci curvature at x (viewed as an element of \hbox{Hom}( TM, TM ) are \lambda+\mu, \mu+\nu, \nu+\lambda. Conclude in particular that

\|\hbox{Riem}\|_g = O( \|\hbox{Ric}\|_g ) (31)

where we endow the (fibres of the) spaces \hbox{Hom}(\bigwedge^2 TM, \bigwedge^2 TM) and \hbox{Hom}( TM, TM ) with the Hilbert (or Hilbert-Schmidt) structure induced by the metric g. \diamond

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