Let $M$ be a manifold and $g$ a metric on $M$. Let $TM$ denote the tangent bundle of $M$, and denote points in $TM$ by $(x,v)$ where $v \in T_xM$.

The Levi-Civita connection of $(M,g)$ induces a splitting of the double tangent bundle $TTM = V \oplus H$, where $V$ is the *vertical distribution*, defined by $V_{(x,v)} = T_{(x,v)}T_xM$ (i.e. the tangent space to the fibre), and $H_{(x,v)}$ is the *horizontal distribution*, which is determined by the connection.

Suppose $A:TM \rightarrow TM$ is a map such that $A(x,v)\in T_xM$ (so the map $A(x,\cdot)$ is a map from $T_xM$ to itself for all $x \in M$).

How does one use the splitting described above to define "partial derivatives" $\nabla_xA$ and $\nabla_vA$, which should be maps:

$(\nabla_xA)(x,v):T_xM \rightarrow T_xM$,

$(\nabla_vA)(x,v):T_xM \rightarrow T_xM$.

These should have the property that if $\gamma(t)$ is a curve on $M$ and $u(t)$ is a vector field along $\gamma$ (so $u(t) \in T_{\gamma(t)}M$ for all $t$), and $\nabla_t$ denotes the covariant derivative along $\gamma$, then

$\nabla_t(A(\gamma,u)) = (\nabla_xA)(\gamma,u) \cdot \dot{\gamma} + (\nabla_vA)(\gamma,u) \cdot \nabla_t u$

(here on the LHS, $A(\gamma,u)$ is itself a vector field along $\gamma$, so the notation $\nabla_t(A(\gamma,u))$ is meaningful).

The expression above "makes sense" intuitively, but I can't get the formalism to work properly.