How to calculate the second fundamental form of a submanifold?

[Alico]

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.

 

[jvicens]

Hello,

Thank you for your question. It is possible to calculate the second fundamental form of a circle as the boundary of a spherical cap without parametrizing the manifolds. However, using a parametrization can make the calculations easier and more straightforward.

To calculate the second fundamental form, you need to first find the unit normal vector to the boundary. This can be done by taking the gradient of the function that defines the boundary. In this case, the function is $f(u,v)=r-h$, where $r$ is the radius of the sphere and $h$ is the height of the spherical cap. Therefore, the unit normal vector is given by $N=\frac{\nabla f}{|\nabla f|}=\frac{\partial f}{\partial u}\hat{u}+\frac{\partial f}{\partial v}\hat{v}$.

Next, you need to find the first fundamental form, which is the metric tensor on the boundary. This can be done by using the parametrization $X(u,v)$ and calculating the dot product of the tangent vectors, $g_{ij}=\langle \frac{\partial X}{\partial u}, \frac{\partial X}{\partial u}\rangle$ and $g_{ij}=\langle \frac{\partial X}{\partial v}, \frac{\partial X}{\partial v}\rangle$. In this case, the first fundamental form will be the same as that of a sphere, since the parametrization is the same.

Finally, you can calculate the second fundamental form using the formula $II=\langle dX, dN\rangle$. This will give you a non-zero result, as the unit normal vector is not constant along the boundary.

If you are getting a zero result, it is possible that there is an error in your calculations or in the parametrization you are using. I would recommend double-checking your work and possibly using a different parametrization to see if you get a different result.

I hope this helps. Let me know if you have any further questions.