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I'm working on the following question:

Suppose we are given an orthonormal set of vector fields $\lbrace X_1, X_2,\ldots, X_n\rbrace$ on an $n$-dimensional Riemannian manifold $(M, g)$. Suppose further that $$ \nabla_{X_i}X_j = \frac{1}{2}[X_1,X_2],\quad i,j\leq n. $$ Show that the Riemann curvature tensor satisfies $$ R(X_i,X_j,X_k,X_l)=-\frac{1}{4}g([[X_i,X_j],X_k],X_l),\quad i,j,k,l\leq n. $$

And I'm quite stumped. Here's what I know:

  1. Since $[X_i,X_j] = \nabla_{X_i}X_j - \nabla_{X_j}X_i$, $\nabla_{X_i}X_j = [X_i,X_j]/2\implies \nabla_{X_i}X_j = - \nabla_{X_j}X_i$.
  2. In this case, since we're told that the given set of vector fields are orthonormal, the Koszul formula gives $$ 2g(\nabla_{X_i}X_j,X_k) = -g(X_i,[X_j,X_k]) + g(X_j,[X_k,X_i]) + g(X_k,[X_i,X_j]), $$ and since $g(X_k,[X_i,X_j]) = 2g(\nabla_{X_i}X_j,X_k)$, this reduces to $g(X_i,[X_j,X_k]) = g(X_j,[X_k,X_i])$, so any cyclic permutation of $i$, $j$, and $k$, in the expression $g(X_i,[X_j,X_k])$ leaves the corresponding value of this expression unchanged. That is: $$ g(X_i,[X_j,X_k]) = g(X_k,[X_k,X_i]) = g(X_k,[X_i,X_j]),\quad i,j,k\leq n. $$
  3. Since, in general, $[X,Y] = \nabla_XY - \nabla_YX$, we also have that $$ -\nabla_{[X_i,X_j]}X_k = -\nabla_{X_k}[X_i,X_j] - [[X_i,X_j],X_k], $$ from which it follows that $$ R(X_i,X_j,X_k,X_l) = \frac{1}{2}g(\nabla_{X_i}[X_j,X_k] + \nabla_{X_j}[X_k,X_i] - 2\nabla_{X_k}[X_i,X_j],X_l) - g([[X_i,X_j],X_k],X_l), $$
  4. The above point leads me to believe that $$ \frac{1}{2}g(\nabla_{X_i}[X_j,X_k] + \nabla_{X_j}[X_k,X_i] - 2\nabla_{X_k}[X_i,X_j],X_l) = -3R(X_i,X_j,X_k,X_l), $$ which, after some fiddling, is equivalent to the assertion that \begin{equation}\label{eq:1} 2\nabla_{X_i}[X_j,X_k]+2\nabla_{X_j}[X_k,X_i]-\nabla_{X_k}[X_i,X_j]-3\nabla_{[X_i,X_j]}X_k = 0 \end{equation} and, to me, this screams Bianchi identity, although evidently I've not been able to make much progress beyond this point. I'm sure the rest of this question has something to do with an explicit evaluation of $\nabla_{X_i}[X_j,X_k]$, but I don't know how to arrive at such an evaluation.
  5. As an additional point, it can be shown that the assertion made in the above point can be re-written as $$ R(X_i,X_j)X_k = -\frac{1}{4}[[X_i,X_j],X_k], $$ which would complete the question, but of course we can't assume that the initial assertion holds, as this is what is to be shown.

I'd like to have a go at (what's left of) this question mostly on my own, so if I could be guided in the right direction (as opposed to being given the solution, that is), perhaps with a referral to the facts I'm meant to be relying on in order to show that this relationship holds, for example, that'd be greatly appreciated.

Thanks in advance!

NOTE In the interest of clarity, I want to remark that, unlike what the notation suggests, the set $\lbrace X_1,X_2,\ldots, X_n\rbrace$ is not comprised of coordinate vector fields. Speaking of, I know the notation I’ve used here is quite unique (in comparison to other questions of this nature I've come across on this site, at least); it's how I've been taught to denote things.

joshuaheckroodt
  • 2,789
  • #1 looks wrong to me. – Deane Apr 28 '23 at 14:14
  • @Deane $2\nabla_{X_i}X_j=[X_i,X_j]=\nabla_{X_i}X_j-\nabla_{X_j}X_i$. Subtract $\nabla_{X_i}X_j$ from both sides to obtain the expression. – Chris Apr 28 '23 at 14:20
  • How did you get the first equality? – Deane Apr 28 '23 at 14:22
  • @Deane from the problem statement, which states that these vector fields satisfy $\nabla_{X_i}X_j=\frac{1}{2}[X_i,X_j]$ – Chris Apr 28 '23 at 14:25
  • This is just a shot in the dark, but if you start with $R(X_i,X_j)X_k=\nabla_{X_i}\nabla_{X_j}X_k-\nabla_{X_j}\nabla_{X_i}X_k-\nabla_{[X_i,X_j]}X_k$, you can use the expressions you derived to rewrite this in terms of single Levi-Civita connections, and Lie brackets. Apply $g(R(X_i,X_j)X_k,X_l)$, and maybe successively using Koszul's formula on each Levi-Civita connection will yield the desired result. – Chris Apr 28 '23 at 14:33
  • @Chris If I understand your comment correctly, you're describing #3, since this is the original problem of evaluating $R(X_i,X_j,X_k,X_l)$, just written in terms of first-order applications of the LC connection, and Lie brackets. It's certainly a good idea, the only issue is that I can't seem to "pry" the LC connections and Lie brackets apart from one another. If we were able to do this, we'd in turn be able to explicitly evaluate $\nabla_{X_i}[X_j,X_k]$, which, I suspect, is the key to solving this problem, but alas... – joshuaheckroodt Apr 28 '23 at 15:54
  • @joshuaheckroodt what I’m saying is to use bilinearity of the metric to separate each term, and then use the Kossul formula on each term. Perhaps you already did that, but I didn’t see it in your calculations – Chris Apr 28 '23 at 16:07

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