# geodesic triangle

Let $M$ be a differentiable manifold (at least two times differentiable) and $A,B,C\in M$ (not necessarily distinct). Let $x_{1},x_{2},x_{3}\in[0,\infty)$. Let $\gamma_{1}\colon[0,x_{1}]\to M$, $\gamma_{2}\colon[0,x_{2}]\to M$, and $\gamma_{3}\colon[0,x_{3}]\to M$ be geodesics such that all of the following hold:

• $\gamma_{1}(0)=A$;

• $\gamma_{1}(x_{1})=B$;

• $\gamma_{2}(0)=A$;

• $\gamma_{2}(x_{2})=C$;

• $\gamma_{3}(0)=B$;

• $\gamma_{3}(x_{3})=C$.

Then the figure determined by $\gamma_{1}$, $\gamma_{2}$, and $\gamma_{3}$ is a geodesic triangle.

Note that a geodesic triangle need not be a triangle. For example, in $\mathbb{R}^{2}$, if $A=(0,0)$, $B=(1,2)$, and $C=(3,6)$, then the geodesic triangle determined by $A$, $B$, and $C$ is $\{(x,2x):x\in[0,3]\}$, which is not a triangle.

geodesic metric space (http://planetmath.org/GeodesicMetricSpace)

Title geodesic triangle GeodesicTriangle 2013-03-22 17:11:31 2013-03-22 17:11:31 Wkbj79 (1863) Wkbj79 (1863) 7 Wkbj79 (1863) Definition msc 53C22