The 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 T_{p}M} , the entire tangent space at p {\displaystyle p} . The Hopf–Rinow theorem gives alternative characterizations of completeness. Let ( M , g )...

R {\displaystyle \mathbb {R} } . The Hopf–Rinow theorem characterizes geodesically complete manifolds. Theorem: Let ( M , g ) {\displaystyle (M,g)} be...

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...

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...

this minimizing sequence need not converge to a geodesic. The metric Hopf-Rinow theorem provides situations where a length space is automatically a geodesic...

compactness theorem (symplectic topology) Gromov–Ruh theorem (differential geometry) Hilbert's theorem (differential geometry) Hopf–Rinow theorem (differential...

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

is compact but not complete, a combination of properties that the Hopf–Rinow theorem disallows for Riemannian manifolds. Causality conditions Globally...

{\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...

groups. It also covers holonomy, the de Rham decomposition theorem and the Hopf–Rinow theorem. According to the review of James Eells, it has a "fine expositional...

compact Riemannian manifold is also geodesically complete (by the Hopf–Rinow theorem), this space shows that the same implication does not generalize to...

of the Ehresmann curvature and nonlinear covariant derivative. By Hopf–Rinow theorem there always exist length minimizing curves (at least in small enough...

Collapsing manifold Complete manifold According to the Riemannian Hopf-Rinow theorem, a Riemannian manifold is complete as a metric space, if and only...

Gauss–Bonnet theorem Hopf–Rinow theorem Cartan–Hadamard theorem Myers theorem Rauch comparison theorem Morse index theorem Synge theorem Weinstein theorem Toponogov...

complete metric space with approximate midpoints is a length space. The Hopf–Rinow theorem states that if a length space ( M , d ) {\displaystyle (M,d)} is complete...

space to a manifold is a kind of "linearization" of the manifold. The Hopf–Rinow theorem asserts that it is possible to define the exponential map on the whole...

metric d(x,y) = | x – y | /(1 + | x – y |). On the other hand, by the Hopf–Rinow theorem for metric spaces, if X is complete, locally compact and geodesic—every...

{\displaystyle X} is actually a proper geodesic space (see metric Hopf-Rinow theorem), and G {\displaystyle G} is finitely generated (see Švarc-Milnor...

is defined. So, if M {\displaystyle M} is complete, then, by the Hopf–Rinow theorem, exp p {\displaystyle \exp _{p}} is defined on the whole tangent space...

to Dr Phil at Friedrich Wilhelms University with doctoral advisors Heinz Hopf and Erhard Schmidt. Her thesis titled: Eine elementargeometrische Eigenschaft...