Category:Definitions/Riemannian Metrics

This category contains definitions related to Riemannian Metrics.
Related results can be found in Category:Riemannian Metrics.

Let $M$ be a smooth manifold.

Let $p \in M$ be a point in $M$.

Let $T_p M$ be the tangent space of $M$ at $p$ with the inner product $\innerprod \cdot \cdot_p$.

Let $g \in \map {\TT^2} M$ be a smooth covariant 2-tensor field such that for all $p$ its value at $p$ is equal to $\innerprod \cdot \cdot_p$:

$\forall p \in M : g_p = \innerprod \cdot \cdot_p$

Then $g$ is known as a Riemannian metric on $M$.