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...

{\displaystyle (M,g)} be a connected and continuous Riemannian manifold. The Hopf–Rinow theorem, in this setting, says that (Gromov 1999) if the metric space ( M...

conclusion of the theorem says, in particular, that the diameter of ( M , g ) {\displaystyle (M,g)} is finite. The Hopf-Rinow theorem therefore implies...

complete are called geodesic manifolds; completeness follows from the Hopf–Rinow theorem. Every compact metric space is complete, though complete spaces need...

the Euler characteristic of the manifold. This theorem is now called the Poincaré–Hopf theorem. Hopf spent the year after his doctorate at the University...

differential point of view Differential geometry of surfaces Geodesic circle Hopf–Rinow theorem – Gives equivalent statements about the geodesic completeness of Riemannian...

completeness are equivalent for these spaces. This is the content of the Hopf–Rinow theorem. A simple example of a non-complete manifold is given by the punctured...

is compact but not complete, a combination of properties that the Hopf–Rinow theorem disallows for Riemannian manifolds. A Lorentzian manifold is an important...

{\displaystyle \mathbb {R} ^{n}.} Furthermore it follows from the Hopf–Rinow theorem that every pairs of points in a Cartan–Hadamard manifold may be connected...

University of Greifswald. He retired in 1972. The Hopf–Rinow theorem is named after Hopf and Rinow. In 1959, he became the director of the Institute for...