Definition:Vertical Tangent Space

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Let $M, \tilde M$ be smooth manifolds.

Let $\tilde g$ be a Riemannian metric on $\tilde M$.

Let $\pi : \tilde M \to M$ be a smooth submersion.

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

The vertical tangent space at $x$, denoted by $V_x$, is defined as the tangent space to the fiber containing $x$:

$V_x := \ker d \pi_x = \map {T_x} {\tilde M_{\map \pi x}}$