- #1
mnb96
- 715
- 5
Hello,
I read that when a manifold has a flat metric, the geodesics are always straight lines in the parameter space. I have two questions:
(1)
If we are given a Clifford torus [tex]S^1 \times S^1[/tex] (which is flat), how do we compute the geodesic between two points? Is the following correct?
[tex]\sqrt{\min\left\{ \left|\theta_{1}-\theta_{2}\right|,\,2\pi-\left|\theta_{1}-\theta_{2}\right|\right\}^2 +\min\left\{ \left|\phi_{1}-\phi_{2}\right|,\,2\pi-\left|\phi_{1}-\phi_{2}\right|\right\}^2}[/tex](2)
Let's consider a torus in [itex]\mathbb{R}^3[/itex] with R=r=1, where R and r are respectively the distance from the center, and the cross-section radius.
Now the geodesics are not straight lines in the parameter space, however can we say that the above formula defines a distance between points on the torus?
If the answer is yes, shall we conclude that, when simply looking for a 'distance', knowing whether a manifold is flat or not is irrelevant?
Thanks!
I read that when a manifold has a flat metric, the geodesics are always straight lines in the parameter space. I have two questions:
(1)
If we are given a Clifford torus [tex]S^1 \times S^1[/tex] (which is flat), how do we compute the geodesic between two points? Is the following correct?
[tex]\sqrt{\min\left\{ \left|\theta_{1}-\theta_{2}\right|,\,2\pi-\left|\theta_{1}-\theta_{2}\right|\right\}^2 +\min\left\{ \left|\phi_{1}-\phi_{2}\right|,\,2\pi-\left|\phi_{1}-\phi_{2}\right|\right\}^2}[/tex](2)
Let's consider a torus in [itex]\mathbb{R}^3[/itex] with R=r=1, where R and r are respectively the distance from the center, and the cross-section radius.
Now the geodesics are not straight lines in the parameter space, however can we say that the above formula defines a distance between points on the torus?
If the answer is yes, shall we conclude that, when simply looking for a 'distance', knowing whether a manifold is flat or not is irrelevant?
Thanks!
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