# Local Convexity and Curvature (Do Carmo)

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!

• What is a "strictly locally convex surface"? Commented Apr 26, 2018 at 4:00
• Sorry, I should have given more definitions. added in edit. Commented Apr 26, 2018 at 4:11
• Well, it would help understand your question and increase the probability of an answer if you added the definition. Cheers! Commented Apr 26, 2018 at 4:13
• A small enough neighbourhood $V$ can be written as a graph of some function $f$ over $T_p S,$ after which the curvatures are proportional to the second partial derivatives of $f$ and lying in one of the half-spaces is equivalent to $f$ having constant sign. From there it's just calculus. Commented Apr 26, 2018 at 5:50
• I don't really follow your proposed example. The fact to be used is the multivariable generalization of the following: if $f \in C^2(\mathbb R)$ satisfies $f(0)=0, f'(0)=0, f''(0)>0,$ then $f(x)>0$ for all $x\ne 0$ in some neighbourhood of $0$. Commented Apr 26, 2018 at 6:52