Definition of locally convex:
(Definition) (Local Convexity and Curvature). A surface $S\subset \mathbb{R}^3$ is locally convex at a point $p\in S$ if there exists a neighborhood $V \subset S$ of $p$ such that $V$ is contained in one of the closed half-spaces determined by $T_p(S) $ in $\mathbb{R}^3$. If, in addition, $V$ has only one common point with $T_p(S)$, then $S$ is called strictly locally convex at $p$.
Consider the surface $S \subset \mathbb{R}^3$ generated by revolving the function $z = x^4$ about the $z$ axis.
Then this surface can be parameterized by $f(x,y) = (x,y,(x^2+y^2)^2)$.
which has zero normal curvature at $(0,0,0)$, in every direction, in particular, the principal direction, so it must be a planar point.
Yet, we know that every point is above the $z$ axis, so it's strictly locally convex.
So strictly locally convexity at a point does not demand that the principal curvature need to be non-zero.
Is this right?
