Does anyone know definition of Levi-Civita connection map that defined as $K: TTM\to TM$. and how to prove the following theorem:



> 
> **Theorem:** If $X\in\mathfrak{X}(M)$ be a vector field over $M$ and $K:TTM\to TM$ Levi-Civita connection map then
>  $$K\circ dX=\nabla X$$
>  where $dX$ is derivation of vector field $X$ as a map.
> 
> 
> 


Thanks.



Definition of Levi-Civita connection map and a theorem about it?
After the comment of Jez, here is the corrected answer: 


There is the well-defined vertical bundle $VTM\subset TTM\to TM$ given as the kernel of the (differential of the) projection.
Moreover, for any $v\in TM,$ there is a natural identification $\Phi$ of $V\_vTM$ with $T\_\pi(v)M.$


Using the Levi-Civita connection (or ans other connection), you can construct a horizontal bundle $HTM\subset TTM\to TM$ with the properties:
(1)
$$HTM\oplus VTM=TTM,$$
$d\pi\colon H\_vTM\to T\_{\pi(v)}M$ is an isomorphism for any $v\in TM$
and (3)
for any vector fields X and Y we have
$$\Phi^{-1} \nabla\_XY= \pi^{VTM} dY(X),$$
where $\pi^{VTM}$ is the projection to $VTM$ corresponding to the splitting (1). The proof is the same as the proof of existence of the geodesic spray. From these properties, you obtain that $K=\Phi\circ \pi^{VTM}$ is satisfying your equation.


