In many References such as *D.E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds* chapter 9, and *Differential Geometric Structures
By Walter A. Poor* Page 54; the horizontal and vertical lift(space) of a vector field on $M$, $X\in\Gamma(TM)$, are defined as follows:

If $X$ is a vector field on $M$, its vertical lift $X^V$ on $TM$ is the vector field defined by $X^V\omega = \omega(X)\circ \pi$, where $\omega$ is a 1-form on $M$, which on the left side of this equation is regarded as a function on $TM$. For an affine connection $\nabla$ on $M$, the horizontal lift $X^H$ of $X$ is defined by $X^H\omega = \nabla_X\omega$.

The span of the horizontal lifts at $t ∈ TM$ is called the horizontal subspace of $T_tTM$.

The other approach is as follows:

It is well-known that the tangent space to $TM$ at $(x, u)$ splits into the direct sum of the vertical subspace $VTM_{(x,u)}=ker\pi_*|_{(x,u)}$ and the horizontal subspace $HTM_{(x,u)}$ with respect to $\nabla$ $$TTM=HTM\oplus VTM.$$ For $X\in T_xM$, there exists a unique vector $X^h$ at the point $(x, u)\in TM$ such that $X^h\in HTM_{(x,u)}$ and $\pi_*(H^h) = X$. $X^h$ is called the horizontal lift of $X$ to $(x, u)$. There is also a unique vector $X^v$ at the point $(x, u)$ such that $X^v\in VTM_{(x,u)}$ and $X.(df) = Xf$ for all functions $f$ on $M$. $X^v$ is called the vertical lift of $X$ to $(x, u)$.

My problems are:

- How can I split every vector field in $TTM$ into horizontal and vertical part?
- What is the geometric interpretation of horizontal and vertical spaces?
- why the tangent sphere bundle and tangent bundle is different in the sense of horizontal and vertical part?

It seems that after solving the question I can to prove the following identities: $$[X^v,Y^v]=0,\quad dX(Y)=Y^h+(\nabla_YX)^v\quad X,Y\in\Gamma(TM).$$

Thanks.