Riemannian distance

Riemannian distance between two points is the infimum of the lengths of piecewise smooth curves joining them, with lengths measured by the manifold’s Riemannian metric.

Riemannian metric: A smoothly varying inner product on each tangent space of a manifold. It assigns the lengths of tangent vectors used to measure curves.
Riemannian manifold: A smooth manifold equipped with a smoothly varying positive-definite inner product on its tangent spaces. Riemannian distance is defined between points of this geometric space.
Metric space: A set equipped with a nonnegative distance function satisfying identity, symmetry, and the triangle inequality. Riemannian distance gives each connected Riemannian manifold this structure.
Intrinsic distance: The infimum of lengths of paths constrained to remain within a given space. Riemannian distance is an intrinsic distance, with lengths supplied by the metric.
Curve length: The integral of a curve’s speed over its parameter interval, measured using a chosen metric. Riemannian distance is the infimum of these lengths over joining curves.
Tangent space: A vector space that represents the possible tangent directions at a point of a smooth manifold. The metric measures curve velocities in these spaces.
Metric completeness: The property that every Cauchy sequence in a metric space converges within that space. Completeness for Riemannian distance is equivalent to geodesic completeness under Hopf–Rinow.
Euclidean distance: The straight-line distance between points in Euclidean space, computed from the standard inner product. On a curved manifold, the shortest path can differ from the ambient straight-line separation.
Geodesic: A curve whose velocity is parallel along itself with respect to a connection. Locally, geodesics realize the shortest lengths between sufficiently close points.
Piecewise smooth curve: A continuous curve that is smooth on each of finitely many subintervals. Such curves provide the paths over which the distance infimum is taken.

Linked from 8 pages