I'm trying to see the equivalence between the Ehresmann connection and the connection map, and having trouble getting the setup to be correct.
Suppose $M$ is a smooth manifold. Let $\pi:TM\to M$ denote tangent bundle, with differential $d\pi:TTM\to TM$. Then one has the canonical vertical subbundle $V=\ker{d\pi}$.
A horizontal subbundle $H$ is any subbundle which is complementary to $V$, that is, $TTM=H\oplus V$. Such an $H$ is equivalent to the existence of some $(0,2)$-tensor $\sigma$ on $TM$ ($\sigma:TTM\to TTM$) such that $\sigma^2=\sigma$ and $\sigma(TTM)=V$, and then letting $H=\ker{\sigma}$.
However, if $(M,g)$ is Riemannian with Levi-Civita connection $\nabla$, from what I've found in the literature, one typically defines a connection map $K:TTM\to TM$ by taking some $(\theta,\xi)\in TTM$, letting $\gamma(t)=(\alpha(t),\beta(t))\in TM$ with $\gamma(0)=\theta$, $\gamma'(0)=\xi$ and giving $$(\theta,\xi)\mapsto K_\theta(\xi)=\left(\nabla_{\alpha'(t)}\beta(t)\right)_{t=0}.$$ Then you define $H=\ker{K}$.
What's the correlation between $\sigma$ and $K$?
Also, if anyone has any references on the above that would be helpful. I seem to have only found a small section Sakai's Riemannian Geometry text, and a bit more in-depth section in Paternain's Geodesic Flows text. My interest in the topic is geared towards understanding the Sasakian metric on $TM$ better.