Let's fix a compact Riemannian manifold $(M,g)$ and a smooth function $f\in C^{\infty}(M)$. The gradient $\nabla f$ is a complete vector field since $M$ is compact so there is an $\mathbb{R}$-action on $M$ given by $$t\cdot p=\phi_t(p) $$ where $\phi_t$ is the one-parameter group of diffeomorphisms generated by $\nabla f$. This induces an action on $TM$ by pushforward $$(t\cdot X)_{\phi_t(p)}={\phi_t}_*(X_p)$$ and an action on $T^*M$ by $$(t\cdot \omega)_{\phi_t(p)}={\phi_{-t}}^*\omega_p $$ These actions in turn obviously induce an action on any tensor products of these bundles.

Can we always find a connection $D:TM\to T^*M\otimes TM$ such that $$t\cdot DX=D(t\cdot X)?$$ What is the relation to the Levi-Civita connection?

(I suppose one could ask more generally about the action coming from any random vector field, but I thought maybe things would be easier in this case.)


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