The most minimal definition of the covariant derivative (or connection) I encountered, i.e. taking at least as possible field information in a neighbourhood, is a map $\nabla: TM \times \Gamma(TM) \to TM$, with $\pi(\nabla_\eta F) = \pi(\eta)$, and other properties (additivity, ...). And of course only the values of $Y$ along a curve $\gamma:I \to M$, with $\gamma'(0) = \eta$, is needed to evaluate $\nabla_\eta F$.
Because of the this answer, where it says that the map $T_{p,v}TM\ni V\mapsto (d\pi(V),\frac{D}{dt}v(0))\in T_pM \oplus T_pM$ is independent of the curve, it seems to me that one can define a connection of only a single variable, namely $\nabla': TTM \to TM$ with the formula for $\frac{D}{dt} Z \equiv \nabla _t Z$, where $Z \in \mathfrak{X}(\gamma)$ is vector field along $\gamma$, by the following way.
Let $\bar{\pi}:TTM \to TM$ and $\pi:TM \to M$ be the natural projections and $$ Y = Y^i \partial_{q^i}|_{\bar{\pi}(Y)} + \dot{Y}^i \partial_{\dot{q}^i}|_{\bar{\pi}(Y)} \in TTM $$ with base point $\bar{\pi}(Y)=X_p \in TM$. In coordinates $\Phi \equiv (x \circ \pi \circ \bar{\pi}, dx \circ \bar{\pi}, dq, d\dot{q})$ where $dx = (dx^1,\dots,dx^m), dq = (dq^1,\dots,dq^m)$ and $d\dot{q}= (d\dot{q}^1,\dots,d\dot{q}^m)$ of $TTM$ the vector $Y$ reads $$ \Phi (Y) = (p^1,\dots,p^m,X^1,\dots,X^m,Y^1,\dots,Y^m,\dot{Y}^1,\dots,\dot{Y}^m). $$ Then, Define $$ \nabla'Y := \left( \dot{Y}^k + X^i Y^j (\Gamma_{ij}^k \circ \pi \circ \bar{\pi} )(Y) \right) \partial_{x^k}|_p $$ Where
- the components $X^i$ play the role of the value of $Z \in \mathfrak{X}(\gamma)$
- the components $Y^i$ play the role of the velocity $\eta$ (of the base point $p$)
- the components $\dot{Y}^i$ play the role of time derivatives of the componnts of $Z$, namely $(Z^i)'$
in the formula $$ \frac{D}{dt} Z := \left( (Z^k)' + \eta^i Z^j (\Gamma_{ij}^k \circ \gamma) \right) \partial_{x^k}|\circ\gamma $$
My question is whether this viewpoint correct and whether this map $\nabla'$ is well-defined ?
