How to calculate the second fundamental form of a submanifold?

  • #1
Alico
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Hi, I'm trying to calculate the second fundamental form of a circle as the boundary (submanifold) of a spherical cap. I'm not sure if I'm doing it right. Is it possible to do that without parametrize the manifolds?

I wrote the parametrization of the spherical cap (which is the same as the sphere, with different range for the angles, $0 < u \leq 2 \pi$ and $0 < v \leq arccos((r-h)/r)$.) and then I took v=arccos((r-h)/r) to get a parametrization, $X$, of the boundary. I wrote the second fundamental form as II = <-dX, dN>, where N is the unity normal to the boundary. I got zero as answer. Is it right?

Is there a way to get a different result (different metric, for example)? How can I do this?

Thank you.
 

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