Due date: Monday 15. April 2024, 10 AM.
Exercise 1
Let \(M\subset \mathbb{R}^3\) be the surface determined by the equation \(z=xy^2.\)
Show that \(F\colon \mathbb{R}^2\rightarrow \mathbb{R}^3\) given by \((u,v)\mapsto (u,v,uv^2)\) is a local parametrization of \(M.\)
Using the parametrization in part (a), determine the first fundamental form of \(M.\)
Using the parametrization in part (a), determine the second fundamental form of \(M.\)
Prove that Gaussian curvature satisfies \(K\le0\) and \(K=0\) only when \(y=0.\)
Prove that \((0,0,0)\) is a planar point of \(M.\)
Exercise 2
Let \(M\subset \mathbb{R}^3\) be the surface determined by the parametrization \(F\colon \mathbb{R}^2\rightarrow \mathbb{R}^3\) such that \((u,v)\mapsto (u,v,u^2-v^2).\)
Determine the first fundamental form of \(M.\)
Determine the Gauss map for \(M.\)
Determine the second fundamental form of \(M.\)
Compute the Gaussian curvature \(K\) and show that every point of \(M\) is hyperbolic.
Exercise 3
Let \(I\subseteq\mathbb{R}\) be an open interval and \(\gamma\colon I\rightarrow \mathbb{R}^3\) an immersed curve. The map \(F\colon I\times\mathbb{R}\rightarrow \mathbb{R}^3\) defined by \((t,v)\mapsto \gamma(t)+v\gamma'(t),\) is called the tangent surface to \(\gamma.\)
Prove that the image of \(F\) is not an immersed surface.
Show that if the curvature \(\kappa\) of \(\gamma\) is nowhere zero, then the image of \(F|_U\colon U\rightarrow\mathbb{R}^3,\) where \(U=\{(t,v)\in I\times\mathbb{R}\ |\ v>0\},\) is an immersed surface.