91Ó°ÊÓ

a. Show that if a curve \(C \subset S\) is both a line of curvature and a geodesic, then \(C\) is a plane curve. b. Show that if a (nonrectilinear) geodesic is a plane curve, then it is a line of curvature. c. Give an example of a line of curvature which is a plane curve and not a geodesic.

Let \(S \subset R^{3}\) be a regular, compact, connected, orientable surface which is not homeomorphic to a sphere. Prove that there are points on \(S\) where the Gaussian curvature is positive, negative, and zero.

Show that if \(\mathbf{x}\) is an isothermal parametrization, that is, \(E=G=\lambda(u, v)\) and \(F=0\), then $$ K=-\frac{1}{2 \lambda} \Delta(\log \lambda) $$ where \(\Delta \varphi\) denotes the Laplacian \(\left(\partial^{2} \varphi / \partial u^{2}\right)+\left(\partial^{2} \varphi / \partial v^{2}\right)\) of the function \(\varphi\). Conclude that when \(E=G=\left(u^{2}+v^{2}+c\right)^{-2}\) and \(F=0\), then \(K=\) const. \(=4 c\).

Compute the Euler-Poincaré characteristic of a. An ellipsoid. b. The surface \(S=\left\\{(x, y, z) \in R^{3} ; x^{2}+y^{10}+z^{6}=1\right\\}\).

Show that in a system of normal coordinates centered in \(p\), all the Christoffel symbols are zero at \(p\).

Consider the torus of revolution generated by rotating the circle $$ (x-a)^{2}+z^{2}=r^{2}, y=0, $$ about the \(z\) axis \((a>r>0)\). The parallels generated by the points \((a+r, 0),(a-r, 0),(a, r)\) are called the maximum parallel, the minimum parallel, and the upper parallel, respectively. Check which of these parallels is a. A geodesic. b. An asymptotic curve. c. A line of curvature.

Show that there exists no surface \(\mathbf{x}(u, v)\) such that \(E=G=1, F=0\) and \(e=1, g=-1, f=0\).

Prove that an orientable compact surface \(S \subset R^{3}\) has a differentiable vector field without singular points if and only if \(S\) is homeomorphic to a torus.

Compute the Christoffel symbols for an open set of the plane a. In Cartesian coordinates. b. In polar coordinates. Use the Gauss formula to compute \(K\) in both cases.

Let \(V\) be a connected neighborhood of a point \(p\) of a surface \(S\), and assume that the parallel transport between any two points of \(V\) does not depend on the curve joining these two points. Prove that the Gaussian curvature of \(V\) is zero.