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.
Also see
- Results about Riemannian metrics can be found here.
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 Über die Hypothesen, welche der Geometrie zu Grunde liegen) to apply for position of Privatdozent (unpaid lecturer) at Göttingen.
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.32$: Riemann ($\text {1826}$ – $\text {1866}$)
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Definitions