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.

This entry is not yet complete, as a geodesic metric space has not yet been defined on PlanetMath. If the words ``geodesic metric space'' are clickable in the previous sentence, please let me know right away. Thanks.