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 $T\_p M$.



Algebraic geometric conditions on the variety $V(F)$ such that the manifold defined by $F$ has nonvanishing second fundamental form?
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 *over*determined 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 $2$.


