Definition:Horizontal Vector Field

Definition
Let $M$ be a smooth manifold.

Let $x \in M$ be a base 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.