Recently I came across the following problem. Here's the setting:
Let $(M^n,g)$ be a Riemannian manifold, $\nabla$ the Levi-Civita connection, and $U$ a coordinate neighbourhood with coordinates $\{x^i\}$. $X = X^i \frac{\partial}{\partial x^i}$ is a vector field. We denote by $\nabla_i$ the derivative $\nabla_\frac{\partial}{\partial x^i}$. Denote also $X^i_j := \nabla_j X^i$. Consider $\{ X^i, X^j_k\}$ to be unknown functions and show that the system $\begin{align} \nabla_i X^j &= X_i^j \\ \nabla_k X_i^j &= -R_{pki}^j X^j \end{align}$$\begin{align} \nabla_i X^j &= X_i^j \\ \nabla_k X_i^j &= -R_{pki}^j X^p \end{align}$ with conditions $X_{ij} = -X_{ji}$, where $X_{ij} = X_i^kg_{kj}$, is completely integrable iff $M^n$ is of constant curvature. In this case the solution depends on $\frac{n(n+1)}{2}$ arbitrary constants.
The vectors in the solution space would be the Killing vector fields. The part with the number of constants is clear, but I couldn't acquire the curvature condition, no matter how I tried to attack it. Applying Frobenius' theorem didn't do the job for me, so I suppose I am making a mistake somewhere, be it logical, or computational. The problem is taken from Kentaro Yano's book "Integral Formulas in Riemannian Geometry", Marcel Dekker Inc., NY, 1970, p.35, Problem 19. Similar problems (21, 23, 25) follow, concerning different infinitesimal transformations. According to the convention of the book, the Riemannian curvature tensor is defined as $R_{ijk}^l \frac{\partial}{\partial x^l}= R(\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j})\frac{\partial}{\partial x^k}$
