Definition:Riemannian Metric/Interpretation

Riemannian Metric: 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:

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

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

Usually the $g_{i j}$ are taken so as to form a non-zero determinant.

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

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

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Riemannian geometry
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Riemannian geometry