How to solve the system of PDEs defining Killing vectors

Question

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 $R_{ijk}^l \frac{\partial}{\partial x^l}= R(\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j})\frac{\partial}{\partial x^k}$

Answer 1

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.