Definition:Vertical Vector Field

Definition
Let $M$ be a smooth manifold.

Let $x \in M$ be a base point.

Let $V_x$ be the vertical tangent space.

Let $Y$ be a vector field on $M$.

Suppose for each $x \in M$ the value of $Y$ lies in the vertical space of $x$:


 * $\forall x \in M : \valueat Y x \in V_x$

Then $X$ is called a vertical vector field.