Is there a natural connection on $TM$

Question

The Sasaki metric gives a natural way to equip $TM$ with a Riemannian metric in case $M$ is already equipped with a Riemanian metric. Question: Let $M$ be manifold equipped with a connection, is there a known natural way to equip $TM$ with a connection ?

Answer 1

The construction of the Sasaki metric relies on the fact that an affine connection $\nabla$ defines a splitting of $TTM$ into vertical and horizontal subbundles $TTM\cong VTM\oplus HTM$: A vector $v\in TTM$ is vertical if it is tangent to a fiber, and horizontal if it is the derivative of a parallel vector field along a curve. Both of these subbundles are canonically isomorphic to the pullback bundle $\pi_{TM}^*(TM)$: The pullback maps are given by the differential $d\pi_{TM}|_{HTM}$ for $HTM$ and (fiberwise) by the canonical isomorphism of vector spaces $T_vV\cong V$ for $VTM$.

The Sasaki metric uses this "sum of pullbacks" structure to induce a metric on $TTM$, but we can do the same thing with an affine connection, since there is a canonical affine connection induced on pullback bundles as well as Whitney sums.

Answer 2

$TM$ is a smooth manifold. Hence admits a Riemannian structure. Thus you may equip it with the levi civita connection.