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 $V_x$:


 * $T_x M = H_x \oplus V_x$