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Let  be a differentiable manifold (at least two times differentiable) and
 (not necessarily distinct). Let
 . Let
![$ \gamma_1 \colon [0,x_1] \to M$](http://images.planetmath.org:8080/cache/objects/9510/l2h/img4.png "$ \gamma_1 \colon [0,x_1] \to M$") ,
![$ \gamma_2 \colon [0,x_2] \to M$](http://images.planetmath.org:8080/cache/objects/9510/l2h/img5.png "$ \gamma_2 \colon [0,x_2] \to M$") , and
![$ \gamma_3 \colon [0,x_3] \to M$](http://images.planetmath.org:8080/cache/objects/9510/l2h/img6.png "$ \gamma_3 \colon [0,x_3] \to M$") be geodesics such that all of the following hold:
Then the figure determined by  ,  , and  is a geodesic triangle.
Note that a geodesic triangle need not be a triangle. For example, in
 , if  ,  , and  , then the geodesic triangle determined by  ,  , and  is
![$ \{(x,2x): x\in[0,3]\}$](http://images.planetmath.org:8080/cache/objects/9510/l2h/img23.png "$ \{(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.
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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
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