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.
Best Answer
I find the following viewpoint helpful to translate between the different incarnation of a connection.
To every vector bundle $\pi: E \to M$ (in your case $E = TM$) we have an associated exact sequence of vector bundles (sometimes called the Atiyah sequence, at least in the principal bundle case): $$ 0 \to V E \to TE \to \pi^* TM \to 0 $$ Here $VE$ denotes the bundle of vertical tangent bundles. Now there are three ways to split an exact sequence:
So now it should be pretty clear how to translate between the different viewpoints (modulo some natural isomorphisms). For example, every tangent vector $X \in T_e E$ can be written as a sum $X^v + X^h$, where $X^v = \omega(X)$ and $X^h$ is the horizontal lift of some vector $Y \in T_{\pi(e)}M$.