Riemannian metric

A smoothly varying positive-definite inner product on the tangent spaces of a smooth manifold. It defines lengths, angles, volumes, and distances on the manifold.

Tangent space: The vector space of directions tangent to a manifold at a specified point. The metric assigns an inner product separately to each tangent space.
Smooth manifold: A space locally modeled on Euclidean space with smoothly compatible coordinate charts. A Riemannian metric equips this smooth space with geometric measurements.
Laplace–Beltrami operator: The differential operator on functions or tensors defined using a Riemannian metric and its Levi-Civita connection. The metric extends the familiar Laplacian to curved spaces.
Pseudo-Riemannian metric: A smoothly varying nondegenerate symmetric bilinear form on tangent spaces that may be indefinite. Unlike a Riemannian metric, it permits both positive and negative squared lengths.
Riemannian length: The length of a curve obtained by integrating the metric's norm of its velocity. Curve lengths arise by measuring each tangent vector with the metric.
Differential form: A smooth, alternating multilinear function on tangent vectors, varying from point to point. The metric's volume form is a differential form built from its inner products.
Hodge star: A linear operator on differential forms determined by a metric and an orientation. It converts the metric's inner products and volume into operations on forms.
Finsler geometry: A geometry in which each tangent space has a smoothly varying, generally non-quadratic norm. It defines lengths without requiring norms to come from inner products.
Geodesic: A curve that locally extremizes length, or equivalently has zero covariant acceleration. The metric determines which curves count as locally straight.
Inner product: A symmetric positive-definite bilinear form that assigns lengths and angles to vectors. A Riemannian metric is a smoothly varying family of inner products.

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