# Definition:Horizontal Tangent Space

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## Definition

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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## Sources

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