Riemannian manifold

A smooth manifold equipped with a smoothly varying positive-definite inner product on each tangent space. This metric defines lengths, angles, volumes, and distances.

Smooth manifold: A space that locally resembles Euclidean space and has smoothly compatible coordinate charts. It supplies the underlying space on which the Riemannian metric is defined.
Riemannian distance: The infimum of lengths of piecewise smooth curves joining two points in a Riemannian manifold. It gives the manifold an intrinsic distance derived from its metric.
General relativity: A theory of gravitation in which spacetime geometry is determined by matter and energy. Its spacetime metric is generally Lorentzian rather than positive-definite.
Pseudo-Riemannian manifold: A smooth manifold with a smoothly varying nondegenerate metric of indefinite signature. Unlike a Riemannian metric, its inner products can assign negative squared lengths.
Euclidean plane: The two-dimensional real plane with its standard flat metric. It is the simplest complete example of a Riemannian manifold.
Tangent space: A vector space of directions tangent to a manifold at a specified point. The metric assigns an inner product separately to each point’s tangent space.
Geodesic: A curve whose velocity is parallel along itself with respect to a connection. Geodesics generalize straight lines and locally extremize curve length.
Riemannian optimization: Optimization methods that use the geometry of a Riemannian manifold as the search space. The metric defines gradients and update directions when parameters have manifold constraints.
Finsler manifold: A smooth manifold whose tangent vectors carry smoothly varying, positively homogeneous norms. Its lengths need not arise from inner products, unlike Riemannian lengths.
Sphere: The set of points at a fixed distance from a center in Euclidean space. Its induced metric gives a standard example of constant positive curvature.

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