Exercise 1. Show that the map endows the space of vector fields with the structure of an abstract Lie algebra. Also establish the Leibniz rule
for all and .
Exercise 2. Let be two connections on TM. Show that there exists a unique rank (1,2) tensor such that
for all vector fields . Now interpret the Christoffel symbol of a connection on TM relative to a frame as the difference of that connection with the flat connection induced by the trivialisation of the tangent bundle induced by that frame.
Exercise 3. Show that is torsion-free if and only if
for all vector fields X, Y (or in coordinate-free notation, ).
Exercise 4. If is a torsion-free connection on TM, and is the tensor form of the curvature R, defined by requiring that
then show that
for all vector fields . What is the analogue of (18) if is replaced by a rank (k,l) tensor?
Exercise 5. Prove this theorem. (Hint: one can either (a) use abstract index notation and study expressions such as , (b) use coordinate-free notation and study expressions such as , or (c) use local coordinates (e.g. use a frame arising from a chart as in Example 2) and work with the Christoffel symbols . 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.
Exercise 6. Show that the above three symmetries of imply that is a self-adjoint section of , and that these conditions are in fact equivalent in three and fewer dimensions. (The claim fails in four and higher dimensions; see comments.)
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 being flat.)
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 ) are . Show that the eigenvalues of the Ricci curvature at x (viewed as an element of are . Conclude in particular that
where we endow the (fibres of the) spaces and with the Hilbert (or Hilbert-Schmidt) structure induced by the metric g.