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.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Methods for Constructing Riemannian Metrics