Algebraic geometric conditions on the variety $V(F)$ such that the manifold defined by $F$ has nonvanishing second fundamental form?

Question

Let $F$ be a homogeneous from in $\mathbb{R}[x_0, .., x_n]$. Then $F$ defines a projective variety $X \subset \mathbb{P}_{\mathbb{C}}^n$. Assume $X$ is smooth. In this case $F=0$ also defines a submanifold $M = \{ \mathbf{x} \in \mathbb{R}^{n+1} \backslash \{ \mathbf{0} \} : F(x_0, .., x_{n+1}) = 0 \}$ of $\mathbb{R}^{n+1}$.

I was wondering if there were conditions I can impose on $F$ or $X$ such that $M$ has nowhere vanishing second fundamental form? Any comments are appreciated!

Edit: by nowhere vanishing second fundamental form I mean given any $p \in M$ the second fundamental form at $p$ does not vanish identically on $T_p M$.

Answer 1

As we know, the projective hypersurface in $\mathbb{P}^n$ defined by a homogeneous polynomial equation $$ F(x^0,\ldots,x^n)=0 $$ of degree $m$ is nonsingular if $x=0$ is the only solution to the equations $$ 0 = F = \partial_0F = \partial_1F = \cdots = \partial_nF. $$ Because $mF = x^0\,\partial_0F + \cdots + x^n\,\partial_nF$, this is at most $n{+}1$ independent equations for $n{+}1$ unknowns. Generically, they are independent; hence, nonsingularity is a generic condition for a nonzero homogeneous polynomial $F$ (of any degree).

If one assumes that the hypersurface is nonsingular, then the points where the second fundamental form of the hypersurface vanishes are the places where $F=0$ and where the Hessian matrix $\mathrm{Hess}(F) = (\partial_i\partial_jF)$ belongs to the $(n{+}1)$-dimensional space of symmetric matrices of the form $$ \begin{pmatrix}\partial_0F\\\vdots\\ \partial_nF\end{pmatrix} \begin{pmatrix}v_0 & \cdots & v_n\end{pmatrix} +\begin{pmatrix}v_0\\\vdots\\ v_n\end{pmatrix} \begin{pmatrix}\partial_0F & \cdots &\partial_nF\end{pmatrix} $$ for general $v = \begin{pmatrix}v_0 & \cdots & v_n\end{pmatrix}$. These conditions are expressed by the satisfaction of ${n\choose 2}$ linear equations on $\mathrm{Hess}(F)$ (with coefficients that are quadratic in the $\partial_iF$), plus the equation $F=0$, of course. Thus, the places where the second fundamental form vanishes identically are the solutions of an overdetermined system of equations when $n>2$.

When $n=2$, this is only $2$ homogeneous equations for $3$ unknowns, so one expects that the generic smooth projective curve in $\mathbb{P}^2$ will have a finite number of flexes. Indeed, when the ground field is $\mathbb{C}$, then it is well-known that the only nonsingular curves without flexes are (nonsingular) conics.

When $n>2$, this is at least as many equations as unknowns, and, again, for the generic hypersurface, the locus of points where the second fundamental form vanishes identically will be empty.

It turns out that, if one asks for the condition that defines the points where the second fundamental form is merely degenerate (which, when $n>2$, is a weaker condition than vanishing identically), then it turns out that this locus is, generically, of dimension $n{-}2$ and, it seems, generalizing the case of $n=2$, that the only case of a smooth hypersurface when this locus is empty is a nonsingular hypersurface of degree $2$.