Dear all,

Thank you for your time reading this post. I am a student in computer science so this viewpoint of the second fundamental form may be interesting to you.

I would like to understand the second fundamental form of an affine (or projective) variety of dimension $m$ in affine (or projective) space $\mathbb{A}^n$ (or $\mathbb{P}V$). It is a bilinear form from the tangent space to the normal space. So it is naturally identified as a three-way tensor.

My problem is that: is there any geometric meaning of the tensorial viewpoint? In particular, I would like to know if there is some geometric intuition for the tensor rank of this tensor.

Thank you.

Best,

Jimmy Qiao

p.s. The point is mainly to *view the second fundamental form as a three−way tensor*. Especially, will the tensor rank tell us something about the infinitesimal variation of the tangent space in the neighborhood? There is some claim that I would like to see: for some point $p\in X$, if there is an affine space $S\subseteq X$ passing through $p$ (in some neighborhood of $p$) then the tensor rank of $II_p$ is somehow bounded by the codimension of $S$. Thank you again.

The above claim is to generalize the following. Consider a hypersurface $H$. If there is an affine space $S\subseteq X$ passing through $p$ (in some neighborhood of $p$) then the rank of Hessian is bounded by 2 times the codimension of $S$.

This may be a wild conjecture... But thank you all!