# Tangent Space as Orthogonal Direct Sum of Horizontal and Vertical Tangent Spaces

Jump to navigation
Jump to search

## Theorem

Let $M$ be a Riemannian manifold.

Let $x \in M$ be a point.

Let $H_x$ and $V_x$ be a horizontal and vertical tangent space of $M$ at $x$ respectively.

Let $T_x M$ be the tangent space of $M$ at $x$.

Then $T_x M$ decomposes as the orthogonal direct sum of $H_x$ and $V_x$:

- $T_x M = H_x \oplus V_x$

## Proof

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.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 `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Sources

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