## 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. $\diamond$