Why does the Levi-Civita connection commute with pullbacks and pushforwards?

Question

If $i: M \to N$ is an embedding of Riemannian manifolds, I am trying to prove that $\nabla i^* T = i^* \nabla T$ for any covariant tensor $T$ (I use the same letter for the two Levi-Civita connections). I have managed to bring it down to proving that $\nabla _{i_* X} i_* Y = i_* \nabla _X Y$ (where it is understood that I silently extend locally the pushforwards of fields from $M$). This looks trivial, it should be the easiest part of my proof, yet I embarassedly confess that I'm blocked: how to I show this? Do I really use the injectivity of $i$ and of $i_*$?

Answer 1

This is not true. For example, suppose $M=\mathbb S^2$, $N=\mathbb R^3$ and $i\colon M\to N$ is the inclusion map. If you take $T$ to be a covariant tensor field on $\mathbb R^3$ with constant coefficients, then $\nabla T\equiv 0$. But $\nabla(i^*T)$ need not be zero. For example, if $T = dx^1$, then $i^*T$ does not have constant norm and therefore cannot be parallel.