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Suppose I have an endomorphism $J:TM \to TM$ and a connection on M. It is possible to define $\nabla_X J$ by transforming $J$ into a (1,1)-tensor and using the extension of $\nabla$ to tensors. Going back we get an endomorphism $\nabla_X J:TM \to TM$.

Is there a way to define $\nabla_X J:TM \to TM$ directly?

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No, we don't get an endomorphism. – Alexei Averchenko Jul 26 '11 at 17:54
Just to clarify: Alexei is referring to your claim that $\nabla J$ is an endomorphism from $TM$ to itself. That is not true. – Willie Wong Jul 27 '11 at 12:31
You're right. I was trying to say $\nabla_x J$. I'll correct it. – Lucas Kaufmann Jul 28 '11 at 18:10
up vote 3 down vote accepted

As was mentioned in the comments, $\nabla L$ is not a (1,1) tensor. It is actually a (1,2) tensor. However contracting with a vector field $X$ gives us an endomorphism $\nabla_X L$ of $TM$ which is equal to $\nabla_X \circ L - L \circ \nabla_X$. You can check this is consistent with the definition you get when you extend $\nabla$ to $T^*M$ and then to $TM^* \otimes TM$.

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