Hopf–Rinow theorem

Hopf–Rinow theorem is a set of statements about the geodesic completeness of Riemannian manifolds. It is named after Heinz Hopf and his student Willi Rinow, who published it in 1931.^[1]

Statement[]

Let ${\displaystyle (M,g)}$ be a connected Riemannian manifold. Then the following statements are equivalent:

1. The closed and bounded subsets of ${\displaystyle M}$ are compact;
2. ${\displaystyle M}$ is a complete metric space;
3. ${\displaystyle M}$ is geodesically complete; that is, for every ${\displaystyle p\in M,}$ the exponential map exp_p is defined on the entire tangent space ${\displaystyle \operatorname {T} _{p}M.}$

Furthermore, any one of the above implies that given any two points ${\displaystyle p,q\in M,}$ there exists a length minimizing geodesic connecting these two points (geodesics are in general critical points for the length functional, and may or may not be minima).

Variations and generalizations[]

• The Hopf–Rinow theorem is generalized to length-metric spaces the following way:
  • If a length-metric space ${\displaystyle (M,d)}$ is complete and locally compact then any two points in ${\displaystyle M}$ can be connected by a minimizing geodesic, and any bounded closed set in ${\displaystyle M}$ is compact.
• The theorem does not hold in infinite dimensions: (Atkin 1975) showed that two points in an infinite dimensional complete Hilbert manifold need not be connected by a geodesic.^[2]
• The theorem also does not generalize to Lorentzian manifolds: the Clifton–Pohl torus provides an example that is compact but not complete.^[3]

Notes[]

References[]

• Jürgen Jost (28 July 2011). Riemannian Geometry and Geometric Analysis (6th Ed.). Universitext. Springer Science & Business Media. doi:10.1007/978-3-642-21298-7. ISBN 978-3-642-21298-7. See section 1.7.
• Voitsekhovskii, M. I. (2001) [1994], "Hopf-Rinow theorem", Encyclopedia of Mathematics, EMS Press