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Let $X$ be a scheme or manifold and $\nabla: V \rightarrow \Omega^1 \otimes V$

be a connection on a vector bundle $V$ on $X$.

Let $R:=\nabla^2$ denote the curvature homomorphism.

Does it hold that $\nabla R =0$?

How does one show this?

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If your $V$ is $TX$ with $X$ a riemannian manifold, and the connection is the Levi-Cività connection, then there is a well known Bianchi identity which would be silly if $\nabla R$ were zero :) Indeed, in this situation, the condition $\nabla R=0$ is the definition of locally symmetric spaces, and not all riemannian manifolds are locally symmetric. –  Mariano Suárez-Alvarez Oct 26 '11 at 12:21

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