Complete Riemannian space
A Riemannian space with its internal distance function that is complete as a metric space with metric
.
Let be a connected Riemannian space with its Levi-Civita connection, then the following three assertions are equivalent: a)
is complete; b) for each point
the exponential mapping
is defined on all of
(where
is the tangent space to
at
); and c) every closed set
that is bounded with respect to the distance
is compact (the Hopf–Rinow theorem). Consequences are: Any two points
in a complete Riemannian space
can be joined in
by a geodesic of length
; any geodesic is indefinitely extendable.
There is a generalization [2] of this theorem to the case of a space with a non-symmetric distance function.
References
[1] | D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968) |
[2] | S.E. Cohn-Vossen, "Existenz kürzester Wege" Compos. Math. , 3 (1936) pp. 441–452 |
Comments
Let be a point of the Riemannian manifold
. Then
is called geodesically complete at
if
is defined on all of
, and
is geodesically complete if this is the case at all
. For
to be complete (or, equivalently, geodesically complete) it suffices that
is geodesically complete at one point.
References
[a1] | W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German) |
Complete Riemannian space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_Riemannian_space&oldid=11489