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The Hopf–Rinow theorem is a set of statements about the geodesic completeness of
Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
s. It is named after
Heinz Hopf Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of dynamical systems, topology and geometry. Early life and education Hopf was born in Gräbschen, German Empire (now , part of Wrocław, Poland) ...
and his student Willi Rinow, who published it in 1931. Stefan Cohn-Vossen extended part of the Hopf–Rinow theorem to the context of certain types of
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
s.


Statement

Let (M, g) be a connected and smooth Riemannian manifold. Then the following statements are equivalent: # The closed and bounded
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
s of M are
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
; # M is a complete
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
; # M is geodesically complete; that is, for every p \in M, the exponential map exp''p'' is defined on the entire
tangent space In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be ...
\operatorname_p M. Furthermore, any one of the above implies that given any two points p, q \in M, there exists a length minimizing
geodesic In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...
connecting these two points (geodesics are in general critical points for the
length Length is a measure of distance. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...
functional, and may or may not be minima). In the Hopf–Rinow theorem, the first characterization of completeness deals purely with the topology of the manifold and the boundedness of various sets; the second deals with the existence of minimizers to a certain problem in the
calculus of variations The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ...
(namely minimization of the length functional); the third deals with the nature of solutions to a certain system of
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...
s.


Variations and generalizations

* The Hopf–Rinow theorem is generalized to length-metric spaces the following way: ** If a length-metric space is complete and
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...
then any two points can be connected by a minimizing geodesic, and any bounded
closed set In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...
is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
. :In fact these properties characterize completeness for locally compact length-metric spaces. * The theorem does not hold for infinite-dimensional manifolds. The unit sphere in a separable Hilbert space can be endowed with the structure of a Hilbert manifold in such a way that antipodal points cannot be joined by a length-minimizing geodesic. It was later observed that it is not even automatically true that two points are joined by any geodesic, whether minimizing or not. *The theorem also does not generalize to
Lorentzian manifold In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere non-degenerate bilinear form, nondegenerate. This is a generalization of a Riema ...
s: the Clifton–Pohl torus provides an example (diffeomorphic to the two-dimensional torus) that is compact but not complete.


Notes


References

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External links

* * {{DEFAULTSORT:Hopf-Rinow theorem Metric geometry Theorems in Riemannian geometry