I find myself unable to start the following problem in Differential Geometry of Curves and Surfaces by Do Carmo, Section 3.3 Problem 24.a
Edit: (Definition) (Local Convexity and Curvature). A surface $S \subset R^3$ is locally convex at a point p ∈ S if there exists a neighborhood V ⊂ S of p such that V is contained in one of the closed half-spaces determined by Tp(S) in R3. If, in addition, V has only one common point with Tp(S), then S is called strictly locally convex at p.
"Prove that S is strictly locally convex at $p$ if the principal curvatures of $S$ at $p$ are nonzero with the same sign (that is, the Gaussian curvature $K(p)$ satisfies $K(p) > 0$)."
I fail to see why this is strictly locally convex, in fact, I fail to see why this must be locally convex.
What if we had a surface that was generated by revolving about the y axis a curve with infinitely many bumps as x approaches to 0 (with decreasing amplitude to bound its derivative)?, but also somehow makesure that this surface is elliptic at (x,y) = (0,0)?
My guts tell me that this would not be a regular surface, but I am unable to prove it.
I would appreciate any hints!