Geodesic region
A connected set of points on a surface
such that for each point
there exists a disc
with centre at
such that
has one of the following forms: 1)
; 2)
is a semi-disc of the disc; 3)
is a sector of
other than a semi-disc; or 4)
consists of a finite number of sectors
of
with no common points except
.
A point is called a regular interior point in the first case, a regular boundary point in the second, an angular point in the third, and a nodal point in the fourth case. A geodesic region that is compact in itself and has no nodal points is called a normal region. A normal region is either a closed surface or a surface with boundary consisting of a finite number of pairwise non-intersecting Jordan polygons.
A geodesic region may be considered as a metric space by introducing the so-called -distance
between two points
and
(the greatest lower bound of the lengths of all rectifiable curves connecting
and
and completely contained in
). A rectifiable arc in
with ends
is called a
-segment if it is the shortest connection between
and
in
. Single points are considered to be
-segments of length zero. For all points of a
-segment the equation
is valid. A
-ray is a ray inside a geodesic region each partial arc of which is a
-segment. A
-line consists of two rays with no points in common other than the origin, such that each arc contained in the line is a
-segment.
A geodesic region has a total curvature if and only if for any sequence of normal regions exhausting the geodesic region the total curvatures tend to a common value. If the Gaussian curvature of the domain is nowhere negative or if it is nowhere positive, then the domain has a total curvature. If the domain does not have a total curvature, then it is always possible to find an exhausting sequence of normal regions with total curvatures tending to . If the boundary of a complete geodesic region, homeomorphic to a closed half-plane, has only a finite number of angular points and if
are the respective angles measured in the geodesic region, then the inequality
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is valid for the total curvature .
References
[1] | S.E. Cohn-Vossen, "Kürzeste Wege und Totalkrümmung auf Flächen" Compos. Math. , 2 (1935) pp. 69–133 |
Comments
References
[a1] | J. Cheeger, D.G. Ebin, "Comparison theorems in Riemannian geometry" , North-Holland (1975) |
Geodesic region. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Geodesic_region&oldid=33173