Definition:Euclidean Metric/Riemannian Manifold

Definition

Let $x \in \R^n$ be a point.

Let $\tuple {x_1, \ldots, x_n}$ be the standard coordinates.

Let $T_x \R^n$ be the tangent space of $\R^n$ at $x$.

Let $T_x \R^n$ be identified with $\R^n$:

$T_x \R^n \cong \R^n$

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Let $v, w \in T_x \R^n$ be vectors such that:

$\ds v = \sum_{i \mathop = 1}^n v^i \valueat {\partial_i} x$

$\ds w = \sum_{i \mathop = 1}^n w^i \valueat {\partial_i} x$

Let $g$ be a Riemannian metric such that:

$\ds g_x = \innerprod v w_x = \sum_{i \mathop = 1}^n v^i w^i$

Then $g$ is called the Euclidean metric.

Source of Name

This entry was named for Euclid.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Definitions