Riemannian distance on a connected manifold

Depends on

• Riemannian speed and length
• Any two points in a connected smooth manifold can be joined by a piecewise c one curve
• The Cauchy-sequence reals have the least-upper-bound property

Used by

• Closed embedded submanifolds of complete Riemannian manifolds are complete Corollary
• Complete connected Riemannian manifolds are proper length spaces Corollary
• Local formula for distance from the centre of a normal neighbourhood Corollary
• Maximal compact subgroups exist and are conjugate in a connected finite center semisimple Lie group 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
• Positive sectional curvature with no fixed lower bound on a noncompact manifold Counterexample
• H-one Riemannian curves and their half-energy Definition
• A potential perturbation beyond the two-trip distance is invisible before time T Example
• Bonnet-Myers for the round sphere Example
• 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
• Hopf–Rinow on a flat cylinder Example
• Length and distance on the circle Example
• Toponogov comparison on a round sphere Example
• A geodesic stops minimizing exactly at its first conjugate point False statement
• Any two points admit a minimizing geodesic False statement
• Geodesic completeness means compactness False statement
• Riemannian distance is defined by the length of a unique shortest curve 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
• The distance function is smooth on all of m times m 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
• Embedded bands joining two framed spheres exist Lemma
• Finite propagation for the wave equation on a Riemannian manifold Lemma
• Global geodesic-polar coordinates from a boundary point on a simple manifold Lemma
• Local length comparison for a conjugate-free geodesic Lemma
• Logarithmic derivative of the radial volume jacobian is the distance laplacian Lemma
• Minimizing along a geodesic is an initial interval property Lemma
• Radial geodesics from one point reach every point under global exponential domain Lemma
• Relative descending paths with finitely many same-index passages 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
• At a conjugate endpoint the index form is degenerate Proposition

…and 11 more results.

Dependency tree · two levels

13 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)