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

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

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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