Riemannian speed and length

Depends on

• Piecewise c one curve on a manifold
• Pointwise norm and angle from a riemannian metric

Used by

• Local formula for distance from the centre of a normal neighbourhood Corollary
• Riemannian isometries preserve length and distance Corollary
• A complete manifold with zero global injectivity radius Counterexample
• A conjugate point at which there are many geodesics Counterexample
• Antipodal points on a round sphere have many minimizing geodesics Counterexample
• Energy of a piecewise smooth curve Definition
• H-one Riemannian curves and their half-energy Definition
• Riemannian distance on a connected manifold Definition
• Cut locus of a point on a flat circle Example
• Cut locus of a point on a round sphere Example
• Cut locus on a flat rectangular torus from the Dirichlet cell Example
• Distance hessian in euclidean space Example
• Exact-solution correction errors vanish in the Gaussian-beam ray limit Example
• Hopf–Rinow on a flat cylinder Example
• Length and distance on the circle Example
• A geodesic stops minimizing exactly at its first conjugate point False statement
• Any two points admit a minimizing geodesic False statement
• Every geodesic segment is globally length minimizing False statement
• Geodesic completeness means compactness False statement
• The cut locus of a point is always a smooth hypersurface False statement
• The distance from p is smooth on m minus p False statement
• A complete no-return extension of a strictly convex manifold Lemma
• A finite endpoint of a maximal unit-speed geodesic produces a Cauchy curve Lemma
• Arcs joining two points of a connected submanifold avoiding finitely many points Lemma
• Finite propagation for the wave equation on a Riemannian manifold Lemma
• Local length comparison for a conjugate-free geodesic Lemma
• Radial geodesics from one point reach every point under global exponential domain Lemma
• Riemannian length is independent of piecewise c one subdivision Lemma
• Surface geodesic flow, index form, and transport primitive Lemma
• Uniform short-geodesic scale on a compact surface Lemma
• A smooth map with pointwise operator norm at most c is c lipschitz for riemannian distance Proposition
• Affine reparametrization of a geodesic is a geodesic Proposition
• At a conjugate endpoint the index form is degenerate Proposition
• Injectivity radius is the infimum of cut times Proposition
• Length-energy inequality and constant-speed equality case Proposition
• A geodesic does not minimize past its first conjugate point Theorem
• Characterization of a cut point Theorem
• Cut time is positive and continuous Theorem
• First variation formula for length Theorem
• Length minimizers are constant-speed geodesics up to reparametrization Theorem

…and 2 more results.

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• John M. Lee, Introduction to Smooth Manifolds, second edition (standard reference, not scraped)
• Ved Datar, Lectures on Riemannian Geometry, September 2025 (standard reference, not scraped)