$\parbox{0.45\textwidth}{\begin{flushleft}\vspace{-\baselineskip} {\L... ...tet}\\ \textbf{Matematiska Institutionen}\\ Thomas Erlandsson \end{flushleft}}$

DIFFERENTIAL GEOMETRY MN1 FALL 1999

PROBLEM 11

11. Let $\,M\,$ be a Riemann surface with constant Gauss curvature $\,K = K_0.$

a)

Calculate the circumference and area of a circle with radius $\,r.$

b)

Calculate the geodesic curvature $\,k_{g}(r)\,$ of a circle with radius $\,r.\,$ Determine $\,\lim_{r\to\infty}k_{g}(r).$

c)

Let $\,c(t)\,$ be a geodesic and $\,\gamma (t) \,$ a curve whose distance to $\,c(t)\,$ is constant $\,= d.\,$ Calculate the geodesic curvature $\,k_{g}\,$ of the equidistant curve $\,\gamma.$

RESULTS

WARNING! THERE ARE VERY LIKELY MISPRINTS HERE!

a)

$$L(C(r)) = \left\{\begin{array}{rrrr} &\frac{2\pi}{\sqrt{K_0}}... ...qrt{-K_0}}\sinh (r\sqrt{-K_0}), \quad K_0<0. \end{array}\right.$$

$$A(c(r)) = \left\{\begin{array}{rrrr} &\frac{\pi}{K_0}(1 - \co... ...0}(\cosh(r\sqrt{-K_0}) - 1), \quad K_0 < 0. \end{array}\right.$$

b)

$$k_{g} = \left\{\begin{array}{rrrr} &\sqrt{K_0}\cot (r\sqrt{K_... ...rt{-K_0}\coth (r\sqrt{-K_0}),\quad K_0 < 0. \end{array}\right.$$

c)

$$k_{g} = \left\{\begin{array}{rrrr} &-\sqrt{K_0}\tan (d\sqrt{K... ...rt{-K_0}\tanh (d\sqrt{-K_0}),\quad K_0 < 0. \end{array}\right.$$

Tillbaka till Differentialgeometri MN1