Exercise 1. Show that
is consistent with the antisymmetry properties of the Riemann tensor, and with the Bianchi identities, as presented in the previous lecture.
Exercise 2. Verify the derivation of
[Aside: – I wonder if there are more direct derivations of (13) and (15) that do not require one to go through so many computations. One can use (22) and (26) below as consistency checks for these formulae, but this does not quite seem sufficient.]
Exercise 3. If varies smoothly in time (but with static endpoints , show that
where at every point of the curve, is the unit tangent, and is the variation field. (Strictly speaking, one needs to work on the pullback tensor bundles on rather than M in order to make the formula in (17) well defined.)
Exercise 4. Establish the first variation formula
where the infimum ranges over all minimal geodesics from x to y (which in particular determine the unit tangent vector S at x and at y).
Exercise 5. Verify
Then use these formulae to give an alternate derivation of (5) and (8).
Exercise 6. Show that the expression
for the Ricci curvature can be used to imply (13). Conversely, use (13) to recover (41) without performing an excessive amount of explicit computation. (Hint: first show that the Ricci tensor can be crudely expressed as . )