This week we studied curvature and connections, in particular the Levi-Cevita connection on the tangent bundle. It is important to do some calculation to see that our intuition agrees with the formula and calculationd. Many nice things happens for Lie groups

1. Let $M$ be the unit shpere $S^2$, and let $(\theta, \phi)$ be spherical coordinates on it. For $\theta_0 \in (0, \pi)$, let $\gamma=\gamma^{(\theta_0)}$ be the circle with $\theta = \theta_0$ $$ \gamma: [0, 1] \to S^2, \quad t \mapsto (\theta_0, 2 \pi t) $$ Let $u_0 = \d_\phi \in T_{\gamma(0)}S^2$ be a tangent vector to the curve. What is the resulting tangent vector after parallel transport $u_0$ one round along long $\gamma$?

2. Let $M$ be a mobius band of unit radius and width $2\delta$ in $\R^3$, that is, $M$ is the image of the map $$ (0, 2\pi] \times (-\delta, \delta) \mapsto \R^3, (\phi, z) \mapsto ¹⁾ \cos \phi, (1 + z \cos (\phi/2)) \sin \phi, z \cos(\phi/2)) $$