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




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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!



viewing the second fundamental form as a tensor
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.


