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[parent] geodesic triangle (Definition)

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.


\begin{pspicture}(-1,0)(3,6) \rput[a](3,6){.} \psline(0,0)(3,6) \psdots(0,0)(1,2... ... \rput[r](-0.2,0){$A$} \rput[r](0.8,2){$B$} \rput[r](2.8,6){$C$} \end{pspicture}

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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Cross-references: triangle, geodesics, differentiable, differentiable manifold
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This is version 4 of geodesic triangle, born on 2007-06-03, modified 2007-12-06.
Object id is 9510, canonical name is GeodesicTriangle.
Accessed 746 times total.

Classification:
AMS MSC53C22 (Differential geometry :: Global differential geometry :: Geodesics)

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