91Ó°ÊÓ

Determine the asymptotic curves and the lines of curvature of the helicoid \(x=v \cos u, y=v \sin u, z=c u\), and show that its mean curvature is zero.

Prove that a vector field \(w\) defined on a regular surface \(S \subset R^{3}\) is differentiable if and only if it is differentiable as a map \(w: S \rightarrow R^{3}\).

Determine the asymptotic curves of the catenoid $$ \mathbf{x}(u, v)=(\cosh v \cos u, \cosh v \sin u, v) . $$

Determine the asymptotic curves and the lines of curvature of \(z=x y\).

Let \(S\) be a regular surface and let \(C \subset S\) be a regular curve on \(S\), nowhere tangent to an asymptotic direction. Consider the envelope of the family of tangent planes of \(S\) along \(C\). Prove that the direction of the ruling that passes through a point \(p \in C\) is conjugate to the tangent direction of \(C\) at \(p\).

Describe the region of the unit sphere covered by the image of the Gauss map of the following surfaces: a. Paraboloid of revolution \(z=x^{2}+y^{2}\). b. Hyperboloid of revolution \(x^{2}+y^{2}-z^{2}=1\). c. Catenoid \(x^{2}+y^{2}=\cosh ^{2} z\).

(Contact of Curves and Surfaces.) A curve \(C\) and a surface \(S\), which have a common point \(p\), have contact of order \(\geq n(n\) integer \(\geq 1)\) at \(p\) if there exists a curve \(\bar{C} \subset S\) passing through \(p\) such that \(C\) and \(\bar{C}\) have contact of order \(\geq n\) at \(p\). Prove that a. If \(f(x, y, z)=0\) is a representation of a neighborhood of \(p\) in \(S\) and \(\alpha(t)=(x(t), y(t), z(t))\) is a parametrization of \(C\) in \(p\), with \(\alpha(0)=p\), then \(C\) and \(S\) have contact of order \(\geq n\) if and only if $$ f(x(0), y(0), z(0))=0, \quad \frac{d f}{d t}=0, \ldots, \frac{d^{n} f}{d t^{n}}=0 $$ where the derivatives are computed for \(t=0\). b. If a plane has contact of order \(\geq 2\) with a curve \(C\) at \(p\), then this is the osculating plane of \(C\) at \(p\). c. If a sphere has contact of order \(\geq 3\) with a curve \(C\) at \(p\), and \(\alpha(s)\) is a parametrization by arc length of this curve, with \(\alpha(0)=p\), then the center of the sphere is given by $$ \alpha(0)+\frac{1}{k} n+\frac{k^{\prime}}{k^{2} \tau} b \text {. } $$ Such a sphere is called the osculating sphere of \(C\) at \(p\).

(Theorem of Beltrami-Enneper.) Prove that the absolute value of the torsion \(\tau\) at a point of an asymptotic curve, whose curvature is nowhere zero, is given by $$ |\tau|=\sqrt{-K}, $$ where \(K\) is the Gaussian curvature of the surface at the given point.

Let \(\lambda_{1}, \ldots, \lambda_{m}\) be the normal curvatures at \(p \in S\) along directions making angles \(0,2 \pi / m, \ldots,(m-1) 2 \pi / m\) with a principal direction, \(m>2\). Prove that $$ \lambda_{1}+\cdots+\lambda_{m}=m H, $$ where \(H\) is the mean curvature at \(p\).

Determine the umbilical points of the elipsoid $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 $$