Furthermore, any one of the above implies that given any two points there exists a length minimizing
geodesic connecting these two points (geodesics are in general
critical points for the
length functional, and may or may not be minima).
In the HopfâRinow 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 (namely minimization of the length functional); the third deals with the nature of solutions to a certain system of
ordinary differential equations.
In fact these properties characterize completeness for locally compact length-metric spaces.[4]
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.[5] It was later observed that it is not even automatically true that two points are joined by any geodesic, whether minimizing or not.[6]
The theorem also does not generalize to
Lorentzian manifolds: the
CliftonâPohl torus provides an example (diffeomorphic to the two-dimensional torus) that is compact but not complete.[7]
Furthermore, any one of the above implies that given any two points there exists a length minimizing
geodesic connecting these two points (geodesics are in general
critical points for the
length functional, and may or may not be minima).
In the HopfâRinow 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 (namely minimization of the length functional); the third deals with the nature of solutions to a certain system of
ordinary differential equations.
In fact these properties characterize completeness for locally compact length-metric spaces.[4]
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.[5] It was later observed that it is not even automatically true that two points are joined by any geodesic, whether minimizing or not.[6]
The theorem also does not generalize to
Lorentzian manifolds: the
CliftonâPohl torus provides an example (diffeomorphic to the two-dimensional torus) that is compact but not complete.[7]