Take a function $h:M\times TM\rightarrow TM$ on a Riemannian manifold $M$ specified by $w=h(x,y,v)$, with $x,y\in M$, $v\in T_xM$ and $w \in T_yM$. The function $h$ is linear in $v$ and $h(x,x,v)=v$. Assume a Riemannian connection $\nabla$ is fixed in $M$.
Let $x=x(t)$ and $y=y(t)$ be any two smooth curves in $M$, $v_x(t)$ be any tangent vector field along $x(t)$ and define an induced vector field $w_y(t):=h(x(t),y(t),v_x(t))$.
We can calculate the covariant derivative $\nabla_{\dot{y}}w_y$ along the curve $y(t)$ and the covariant derivative $\nabla_{\dot{x}}v_x$. The two covariant derivatives are clearly related to one another. I'm wondering whether the following formula is correct:
$\nabla_{\dot{y}}w_y=(d_xh)_{x,y,v_x}(\dot{x})+(d_yh)_{x,y,v_x}(\dot{y})+h(x,y,\nabla_{\dot{x}}v_x)$
And do the tangent (pushforward) map $(d_xh)_{x,y,v_x}$ enjoy any special property ? For example, I'm wondering whether its `diagonal' part $(d_xh)_{x,x,v_x}$ has any special meaning ?
Thanks a lot in advance for any hints!