Definition:Horizontal 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.

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 $V_x$:

$H_x := \paren {V_x}^\perp$