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!
Best Answer
I believe the definition of the second fundamental form for a projective variety is explained very nicely in
Griffiths, Phillip; Harris, Joseph Algebraic geometry and local differential geometry. Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 3, 355–452.