Why is it that if I have a smooth manifold and a connection map $K$, defined below, is it the case that it induces a decomposition of the tangent space to the tangent space to the manifold, given as the Whitney sum of the kernel of the differential of the projection map ($\pi: TM \to M$, so $d\pi: TTM \to TM$) and the kernel of the differential of $K$?
I was able to understand the definitions, domains, and so forth of many of these objects. However, the proof that ker$(d_v\pi)\oplus$ker$(K_v)=T_vTM$ eludes me.
Theorem. Let $M$ be a differential manifold with an affine connection $\nabla$. Then, there exists a unique differentiable function $K: TTM \longrightarrow TM$, called the connection function, that satisfies the following conditions:
(1) If $v \in TM$ and $\pi(v) = p$, where $\pi: TM \longrightarrow M$ is the canonical projection, then $K(T_v TM) \subseteq T_p M$.
(2) $K_v := K|_{T_v TM}: T_v TM \longrightarrow T_p M$ is a linear map.
(3) If $v \in TM$ and $w \in T_v TM$ are such that $w = d_p X(Y)$, where $Y \in T_p M$ and $X \in {X \in \mathfrak{X}(M): X(p) = v}$, then $K(w) = \nabla_Y X$.
NEW VERSION:
I find this easier to understand if I work it out for an arbitrary vector bundle. I've also omitted details that I find easier to work out if you use a frame of sections.
Let $M$ be an $n$-manifold and $\pi: E \rightarrow M$ be a vector bundle of rank $k$. The differential of $\pi$ is a bundle map, where if $p \in M$ and $e \in E_p$, $$ \pi_*: T_eE \rightarrow T_pM. $$ Since $\pi$ is a submersion, $\pi_*$ is a surjection. The differential of a section $s: M \rightarrow E$ is a bundle map $$ s_*: T_pM \rightarrow T_{s(p)}E, $$ Since $\pi\circ s(p) = p$, it follows that $$ \pi_*\circ s_*: T_pM \rightarrow T_pM $$ is the identity map.
Let $\nabla$ be an affine connection on $E$. For each $p \in M$ and $e \in E_p$, there is a linear map $$ \phi: T_eE \rightarrow E_p $$ such that for any section $s: M \rightarrow E$ and $X \in T_pM$, $$ \nabla_Xs(p) = \phi(s_*X). $$ Conversely, given any $\dot{e} \in E_p$, it is not difficult (say, using a local frame of sections of $E$) that there exists a section $s$ of $E$ such that $s(p) = e$ and $\nabla_Xs(p) = \dot{e}$. Therefore, $\phi$ is surjective and $$ \dim \ker\phi = n. $$ In particular, using parallel translation, given any $X \in T_pM$ and $e \in E_p$, there exists a section $s$ such that $s(p) = e$ and $\nabla_Xs(p) = 0$. Therefore, $s_*X \in \ker \phi$. This defines an injective linear map \begin{align*} T_pM &\rightarrow \ker\phi\\ X &\mapsto s_*X. \end{align*} From this, it follows that $$ \ker \pi_* \cap \ker\phi = \{0\}. $$ Therefore, $$ T_eE = V_p \oplus H_e, $$ where $V_p = \ker \pi_*$ and $H_p = \ker\phi$.