# geodesic completeness

A Riemannian metric on a manifold $M$ is said to be geodesically complete^{} iff its geodesic flow is a complete flow,
i.e. iff for every point $p\in M$ and every tangent vector $v\in {T}_{p}M$ at $p$ the solution to the geodesic equation

$${\nabla}_{\dot{\gamma}}\dot{\gamma}=0$$ |

with initial condition^{} $\gamma (0)=p$, $\dot{\gamma}(0)=v$ is defined for all time.
The Hopf-Rinow theorem
asserts that a Riemannian metric is complete if and only if the corresponding metric on $M$ defined by

$$d(p,q):=inf\{L(c),c:[0,1]\to M,c(0)=p,c(1)=q\}$$ |

is a complete metric (i.e. Cauchy sequences converge). Here $L(c)$ denote the length of the smooth curve $c$, i.e.

$$L(c):={\int}_{0}^{1}{\parallel \dot{c}(t)\parallel}_{c(t)}\mathit{d}t$$ |

For a proof of the Hopf-Rinow theorem see Milnor’s monograph Morse Theory Princeton Annals of Math Studies 51 page 62.

Title | geodesic completeness |
---|---|

Canonical name | GeodesicCompleteness |

Date of creation | 2013-06-03 13:04:01 |

Last modified on | 2013-06-03 13:04:01 |

Owner | jacou (1000048) |

Last modified by | unlord (1) |

Numerical id | 14 |

Author | jacou (1) |

Entry type | Definition |

Classification | msc 53C22 |