Definition:Horizontal Tangent Space

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 $V_x$:

$H_x := \paren {V_x}^\perp$

Further research is required in order to fill out the details. In particular: notation You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by finding out more. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Research}} from the code.

This article needs to be linked to other articles. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{MissingLinks}} from the code.

Sources

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