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Let $M$ be a manifold. Let $\nabla$ be a symmetric connection on $TM$. Denote by $$ d_\nabla:\Omega^1(M;TM)\to\Omega^2(M;TM) $$ the exterior covariant derivative on tangent-vector-valued 1-forms. What can be said about the kernel of $d_\nabla$?

Symmetry implies that $d_\nabla(\operatorname{Id}_{TM})=0$. If $\nabla$ is flat, then for every vector field $X$, $d_\nabla(\nabla X)=0$. What else? In particular, is there a generalization of $d_\nabla(\nabla X)=0$ when the curvature is non-zero?

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