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 $\begin{align} \nabla\_i X^j &= X\_i^j \\ \nabla\_k X\_i^j &= -R\_{pki}^j X^p \end{align}$ 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 $R\_{ijk}^l \frac{\partial}{\partial x^l}= R(\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j})\frac{\partial}{\partial x^k}$



How to solve the system of PDEs defining Killing vectors
The system of PDEs in the original post actually defines a connection $D$ on the bundle $\mathcal{E} = TM \oplus \mathfrak{so}(TM)$, by
$$ D\_X \begin{pmatrix} Y \\ A \end{pmatrix} = \begin{pmatrix} \nabla\_X Y + A(X)\\ \nabla\_X A - R(X,Y) \end{pmatrix} $$
for all vector fields $X,Y$ and field $A$ of skewsymmetric endomorphisms of $TM$. Killing vectors are in bijective correspondence with $D$-parallel sections of $\mathcal{E}$, a result due to Kostant and rediscovered by Geroch, who coined the term *Killing transport*.


Complete integrability is simply the flatness of $D$. The curvature of $D$ is not difficult to calculate and one finds that $D$ is flat if and only if $R(X,Y)Z = \lambda \left( g(Y,Z)X - g(X,Z)Y \right)$, for some $\lambda \in \mathbb{R}$. This is precisely the statement that the metric has constant sectional curvature.


