\begin{definition}[Definition:Horizontal Vector Field]
Let $M$ be a smooth manifold.
Let $x \in M$ be a point.
Let $H_x$ be the horizontal tangent space.
Let $V$ be a vector field on $M$.
Suppose for each $x \in M$ the value of $V$ lies in the horizontal space of $x$:
:$\forall x \in M : \valueat V x \in H_x$
Then $V$ is called a '''horizontal vector field'''.
\end{definition}