Definition:Riemannian Metric

Definition

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 $M$.

Interpretation

Consider a smooth manifold $\MM$ on the real space $\R^n$.

A Riemannian metric on $\MM$ is a metric $\d s$ between nearby points $\tuple {x_1, x_2, \ldots, x_n}$ and $\tuple {x_1 + \d x_1, x_2 + \d x_2, \ldots, x_n + \d x_n}$ by means of the quadratic differential form:

$\ds \d s^2 = \sum_{i, j \mathop = 1}^n g_{i j} \rd x_i \rd x_j$

where each $g_{i j}$ is a suitable real-valued function of $x_1, \ldots, x_n$.

Different instances of $g_{i j}$ define different Riemannian geometries on the manifold under discussion.

A manifold with such a Riemannian metric applied is known as a Riemannian manifold.

Source of Name

This entry was named for Bernhard Riemann.

Historical Note

The concept of a Riemannian metric was originated by Bernhard Riemann in his trial lecture (published as Ueber die Hypothesen, welche der Geometrie zu Grande liegen) to apply for position of Privatdozent (unpaid lecturer) at Göttingen.