viewing the second fundamental form as a tensor

Question

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!

Answer 1

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.

Answer 2

The geometric meaning you are looking for is related to the so called Gauss map.

The Gauss map for a smooth projective variety $X \subset \mathbb{P}^n$ is the rational map

$\mathcal{G} \colon X \to \mathbb{G}(k, n)$

sending the point $p \in X$ to its projective tangent plane $\mathbb{T}_p(X)$.

Such a Gauss map $\mathcal{G}$ induces a differential map on (affine) tangent spaces as follows:

$(d\mathcal{G})_p \colon T_p(X) \to \textrm{Hom}(\mathbb{T}_p(X), K^{n+1}/\mathbb{T}_p(X))$.

Every homomorphism $\phi \colon \mathbb{T}_p(X) \to K^{n+1}/\mathbb{T}_p(X)$ in the image of the Gauss map has $p$ in its kernel, so $(d\mathcal{G})_p$ induces a map

$(d\mathcal{G})_p \colon T_p(X) \to \textrm{Hom}(\mathbb{T}_p(X)/p, K^{n+1}/\mathbb{T}_p(X))$.

Using the identifications

$T_p(X)=\textrm{Hom}(p, \mathbb{T}_p(X)/p)$,

$N_p(X)=T_p(\mathbb{P}^n)/T_p(X)=\textrm{Hom}(p, K^{n+1}/\mathbb{T}_p(X))$

one sees that this is equivalent to a map

$(d\mathcal{G})_p \colon T_p(X) \to \textrm{Hom}(T_p(X), N_p(X))$, i.e. to a bilinear map

$(d\mathcal{G})_p \colon T_p(X) \otimes T_p(X) \to N_p(X)$ inducing

$(d\mathcal{G})_p \colon S^2 T_p(X) \to N_p(X)$

which is exactly the second fundamental form $II_p$.

So, roughly speaking, $II_p$ measures the "variation" of the (projective) tangent space of $X$ in a neighborhood of the point $p$.

You can find more details in the book of Joe Harris "Algebraic geometry: a first course", especially Chapter 17.

EDIT The following result is due to Landsberg (see the paper "On second fundamental forms of projective varieties", Inventiones Mathematicae 117) and deals with the rank of the second fundamental form of a variety containing a linear space. The original statement is more general and also includes the case where $X$ is singular.

THEOREM Let $X^n \subset \mathbb{P}^{n+a}$ be a smooth variety not contained in a hyperplane. Let $L \subset X$ be a $r$-plane. Then for general $p \in L$ we have

$\dim II_p(\underline{L}, T_p(X)) \geq \min \{r,a\}$,

where $\underline{L}$ denotes the tangent directions to $L$ and $II_p(\underline{L}, T_p(X)) = \textrm{Image} \; II_p|_{L \times T_p(X)}$