Difference between revisions of "Hopf-Rinow theorem"
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Revision as of 18:52, 24 March 2012
If is a connected Riemannian space with distance function
and a Levi-Civita connection, then the following assertions are equivalent:
1) is complete;
2) for every point the exponential mapping
is defined on the whole tangent space
;
3) every closed set that is bounded with respect to
is compact.
Corollary:
Any two points can be joined in
by a geodesic of length
. This was established by H. Hopf and W. Rinow [1].
A generalization of the Hopf–Rinow theorem (see [4]) is: If and
are two points in
, then either there exists a curve joining them in a shortest way or there exists a geodesic
emanating from
with the following properties: 1)
is homeomorphic to
; 2) if a sequence of points on
does not have limit points on
, then it does not have limit points in
, that is,
is closed in
; 3)
contains the shortest connection between any two points on
; 4)
for every point
; and 5) the length of
is finite and does not exceed
. Here the function
is not necessarily symmetric, and every point can be joined in a shortest possible (not necessarily unique) way to any point in a certain neighbourhood
.
Corollary:
If there are no bounded rays in , then every bounded set in
is compact.
References
[1] | H. Hopf, W. Rinow, "Ueber den Begriff der vollständigen differentialgeometrischen Flächen" Comm. Math. Helv. , 3 (1931) pp. 209–225 |
[2] | G. de Rham, "Sur la réducibilité d'un espace de Riemann" Comm. Math. Helv. , 26 (1952) pp. 328–344 |
[3] | D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968) |
[4] | S.E. Cohn-Vossen, "Some problems of differential geometry in the large" , Moscow (1959) (In Russian) |
Comments
Let . The manifold
is called geodesically complete at
if
is defined on all of
. The manifold
is geodesically complete if this holds for all
. The Hopf–Rinow theorem also includes the statement that geodesic completeness is equivalent to geodesic completeness at one
.
A geodesic joining and
and of minimal length is called a minimizing geodesic.
References
[a1] | W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German) |
Hopf-Rinow theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hopf-Rinow_theorem&oldid=22591