I am trying to compute the second fundamental form of a hypersurface that is normal to a predefined vector field. Any help would be appreciated.
More specifically, I consider a vector field $\mathbf{v}(\mathbf{x}) \in \mathbb{R}^n$ where $\mathbf{x} \in \mathbb{R}^n$. Next, I fix a point $\mathbf{p} \in \mathbb{R}^n$ and locally construct a surface that is normal to the vector field $\mathbf{v}$. Finally, I want to compute the second fundamental form in terms of $\mathbf{v}$.
In some cases, this is a trivial exercise if one can obtain the equations for the surface that is normal to the vector field $\mathbf{v}$. For example, if $\mathbf{v}(\mathbf{x}) = \mathbf{x}$, the corresponding hypersurface is a hypersphere whose second fundamental form is easily determined. However, is there a way to obtain the second fundamental form by a direct calculation from the vector field $\mathbf{v}$? Also, is there a way to determine whether the second fundamental form is positive-definite (or negative-definite) based on some properties of the vector field $\mathbf{v}$?
Thank you very much for any help.