2009-28.md

course	course_year	question_number	tags	title	year
Geometry	IB	28	IB 2009 Geometry	Paper 4, Section II, G	2009

Let $U \subset \mathbb{R}^{2}$ be an open set. Let $\Sigma \subset \mathbb{R}^{3}$ be a surface locally given as the graph of an infinitely-differentiable function $f: U \rightarrow \mathbb{R}$. Compute the Gaussian curvature of $\Sigma$ in terms of $f$.

Deduce that if $\widehat{\Sigma} \subset \mathbb{R}^{3}$ is a compact surface without boundary, its Gaussian curvature is not everywhere negative.

Give, with brief justification, a compact surface $\widehat{\Sigma} \subset \mathbb{R}^{3}$ without boundary whose Gaussian curvature must change sign.