Geodesic region

A connected set $G$ of points on a surface $F$ such that for each point $x$ there exists a disc $K(x)$ with centre at $x$ such that $K_G=G\cap K(x)$ has one of the following forms: 1) $K_G(x)=K(x)$; 2) $K_G(x)$ is a semi-disc of the disc; 3) $K_G(x)$ is a sector of $K(x)$ other than a semi-disc; or 4) $K_G(x)$ consists of a finite number of sectors $u_i(x)$ of $K(x)$ with no common points except $x$.

A point $x$ 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 $G$-distance $\rho_G$ between two points $a$ and $b$ (the greatest lower bound of the lengths of all rectifiable curves connecting $a$ and $b$ and completely contained in $G$). A rectifiable arc in $G$ with ends $a,b$ is called a $G$-segment if it is the shortest connection between $a$ and $b$ in $G$. Single points are considered to be $G$-segments of length zero. For all points of a $G$-segment the equation $\rho_G(a,x)+\rho_G(x,b)=\rho_G(a,b)$ is valid. A $G$-ray is a ray inside a geodesic region each partial arc of which is a $G$-segment. A $G$-line consists of two rays with no points in common other than the origin, such that each arc contained in the line is a $G$-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 $\pm\infty$. If the boundary of a complete geodesic region, homeomorphic to a closed half-plane, has only a finite number of angular points and if $\omega_1,\dots,\omega_n$ are the respective angles measured in the geodesic region, then the inequality

$$C(G)\leq\pi-\sum_{i=1}^n(\pi-\omega_i)$$

is valid for the total curvature $C(G)$.

References

[1]	S.E. Cohn-Vossen, "Kürzeste Wege und Totalkrümmung auf Flächen" Compos. Math. , 2 (1935) pp. 69–133
[a1]	J. Cheeger, D.G. Ebin, "Comparison theorems in Riemannian geometry" , North-Holland (1975)