\begin{definition}[Definition:Horizontal Tangent Space]
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 point.
Let $V_x$ be the vertical tangent space at $x$.
Then the '''horizontal tangent space at $x$''', denoted by $H_x$, is defined as the orthogonal complement of $V_x$:
:$H_x := \paren {V_x}^\perp$
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\end{definition}