Once we define an Ehresmann connection over the double tangent space, how does this explicitly determine canonically a (Koszul) connection on the base manifold?
Given a smooth manifold $ M $, let $ TM $ be the tangent bundle. The double tangent space is defined as $ T(TM) $.
An Ehresmann connection on a fiber bundle $ \pi: E \to M $ with typical fiber $ F $ is a smooth distribution $ H $ of the tangent bundle $ TE $ which is complementary to the vertical bundle $ VE $, i.e. $ TE = VE \oplus H $, and invariant under the action of the structure group of the bundle.
For the tangent bundle, an Ehresmann connection gives us a way to differentiate vector fields along the directions of other vector fields.
Now, let's focus on the Koszul connection. Given a smooth manifold $ M $ with a Riemannian metric $ g $, a Koszul connection on $ M $ is a map
$$\nabla: \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM) $$
that satisfies the Leibniz rule and metric compatibility with $ g $.
Given the Ehresmann connection over the double tangent space, I am interested in the canonical determination of a Koszul connection on the base manifold.
The question now is how to make this connection between the Ehresmann connection on $ T(TM) $ and the Koszul connection on the base manifold $ M $ explicit.
Any insights on this relationship would be greatly appreciated!