Paper 3, Section II, G

Geometry
Part IB, 2017

Let σ:UR3\sigma: U \rightarrow \mathbb{R}^{3} be a parametrised surface, where UR2U \subset \mathbb{R}^{2} is an open set.

(a) Explain what are the first and second fundamental forms of the surface, and what is its Gaussian curvature. Compute the Gaussian curvature of the hyperboloid σ(x,y)=(x,y,xy)\sigma(x, y)=(x, y, x y).

(b) Let a(x)\mathbf{a}(x) and b(x)\mathbf{b}(x) be parametrised curves in R3\mathbb{R}^{3}, and assume that

σ(x,y)=a(x)+yb(x)\sigma(x, y)=\mathbf{a}(x)+y \mathbf{b}(x)

Find a formula for the first fundamental form, and show that the Gaussian curvature vanishes if and only if

a(b×b)=0\mathbf{a}^{\prime} \cdot\left(\mathbf{b} \times \mathbf{b}^{\prime}\right)=0