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

DIFFERENTIAL GEOMETRY MN1 FALL 1999

EXAMPLES 1-4

1. Let $\,f:U\,\rightarrow\, R ^3\,$ be

$$(u,v) \to (\cos u \cos v, \cos u \sin v, \sin u), \quad (u,v) \,\in \,]- \frac{\pi}{2}, \frac{\pi}{2}[ \,\times\, R.$$

The image $\,f(U)\,$ of $\,U\,$ is then the two-punctured sphere $\,S^2 \backslash \left\{0,0,\pm 1\right\}.$ Then

a) $E = 1, \ \ \ \ F = 0, \ \ \ \ G = \cos^2 u$

b) $n(u,v) = \frac{f_u \times f_v}{\vert f_u \times f_v\vert} = - f(u,v)$

c) $II = - dn \cdot df = df \cdot df = I$

d) $\kappa_1 = \kappa_2 = 1, \ \ \ \ H = K = 1.$ 2. Let $\,f: U \to R ^3\,$ be the torus

$$(u,v) \to ((a + b\cos u)\cos v, (a + b \cos u)\sin v, b \sin u), \quad 0 < b < a,\quad (u,v) \, \in \, R \,\times \, R.$$

Then

a) $f\,$ is regular at each point

b) $E = b^2, \qquad F = 0, \qquad G = (a + b\cos u)^2$

c) $L = b, \qquad M = 0, \qquad N = (a + b \cos u)\cos u$

d) $a^1_1 = \frac{1}{b}, \ \ \ \ a^2_1 = a^1_2 = 0, \ \ \ \ a^2_2 = \frac{\cos u}{a + b \cos u}, \qquad \kappa_1 = a^2_2 < \kappa_2 = a^1_1 = \frac{1}{b}$

e) $K = \frac{\cos u}{b(a + b\cos u)}, \qquad H = \frac{a + 2b\cos u}{2b(a + b\cos u)}.$

Remark $\qquad a^k_i = \sum_j h_{ij}g^{jk}.$

3. A surface $\,f: U \to R ^3\,$ defined by $f(u,v) = (u,v,F(u,v))\,$ can be described as $\,z = F(x,y).$ Then

a)

$$(g_{ij}) = \begin{pmatrix} 1+ F_u^2 &F_uF_v\\ F_uF_v & 1+F_v... ...}} \begin{pmatrix} F_{uu}&F_{uv}\\ F_{uv}&F_{vv} \end{pmatrix}$$

b)

$$2H = \frac{(1+F_u^2)F_{vv} + (1+F_v^2)F_{uu}-2F_uF_vF_{uv}}{(1+F_u^2+F_v^2)^\frac{3}{2}}.$$

4. $f(u,v) = (u,v,u(u^2-3v^2)) \,$ is a so called "monkey saddle". It is a saddle used by a monkey riding a bicycle. There are two depressions for its legs, and an extra one for its tail. Show that the surface has a planar, umbilic point at origo, i.e that $\,K = 0\,$ at origo and that the normal curvatures there are equal in all directions.

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