Hopf-Rinow's theorem

The set of Hopf Rinow is a key message from the Riemann geometry . It says that in the case of Riemannian manifolds, the concepts of geodetic completeness and completeness in the sense of metric spaces coincide. A Riemannian manifold with this property is then called a complete Riemannian manifold . The sentence is named after the mathematician Heinz Hopf and his student Willi Rinow .

Geodetically complete manifold

A connected Riemannian manifold is called geodetically complete if the exponential mapping is defined for all of them . That means, for every point and every tangential vector the geodesic with and on is completely defined. ${\displaystyle (M,g)}$ ${\displaystyle p\in M}$ ${\displaystyle \exp _{p}}$ ${\displaystyle v\in T_{p}M}$ ${\displaystyle p\in M}$ ${\displaystyle v\in T_{p}M}$ ${\displaystyle \gamma }$ ${\displaystyle \gamma (0)=p}$ ${\displaystyle {\dot {\gamma }}(0)=v}$ ${\displaystyle \mathbb {R} }$

Theorem from Hopf and Rinow

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

1. The manifold is geodetically complete. ${\displaystyle M}$
2. There is one such that is defined for all . ${\displaystyle p\in M,}$ ${\displaystyle \exp _{p}}$ ${\displaystyle v\in T_{p}M}$
3. The manifold is complete as metric space . ${\displaystyle M}$
4. The Heine-Borel property applies. That is, every closed and bounded subset is compact .

From these four equivalent statements, another one can be deduced.

• There is a geodesic for all of them , which connects the points and in the shortest possible way. ${\displaystyle p,q\in M}$ ${\displaystyle \gamma }$ ${\displaystyle p}$ ${\displaystyle q}$

The distance function is defined here as the infimum over the arc lengths of all piecewise differentiable curves with and ; that is, it applies ${\displaystyle d(p,q)}$ ${\displaystyle \gamma }$ ${\displaystyle \gamma (a)=p}$ ${\displaystyle \gamma (b)=q}$

${\displaystyle d(p,q)=\inf _{\gamma }\int _{a}^{b}{\sqrt {g_{\gamma (t)}\left({\frac {d\gamma (t)}{dt}},{\frac {d\gamma (t)}{dt}}\right)}}\,{d}t.}$

This distance function makes it a metric space . ${\displaystyle M}$

Corollaries

• From the Hopf-Rinow theorem it follows that all compact, connected Riemannian manifolds are (geodetically) complete.
• For a compact, connected Lie group, it follows that the exponential map is surjective. ${\displaystyle \exp _{G}\colon {\mathfrak {g}}\to \operatorname {G} }$
• All closed submanifolds of a complete, connected Riemannian manifold are complete

Examples

• The sphere , the Euclidean space and the hyperbolic space are complete. ${\displaystyle \mathbb {S} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {H} ^{n}}$
• The metric space with the Euclidean metric induced by the standard scalar product is not complete. If you choose a point , there is no shortest connection to the point . ${\displaystyle M:=\mathbb {R} ^{2}\setminus \{0\}}$ ${\displaystyle p=(x_{1},x_{2})\in M}$ ${\displaystyle q=(-x_{1},-x_{2})\in M}$ ${\displaystyle M}$

literature

• H. Hopf, W. Rinow: About the concept of the complete differential geometric surface . Commentarii Mathematici Helvetici. 3: 209-225, 1931.
• J. Jost: Riemannian Geometry and Geometric Analysis. Springer-Verlag, Berlin 2002, ISBN 3-540-42627-2 .
• Manfredo Perdigão do Carmo: Riemannian Geometry. Birkhäuser, Boston 1992, ISBN 0-8176-3490-8 .