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- | 2. Let $M$ be a mobius | + | 2. Let $M$ be a Mobius |
- | $$ (0, 2\pi] \times (-\delta, \delta) \mapsto | + | $$ (0, 2\pi] \times (-\delta, \delta) \to \R^3$$ |
+ | where | ||
+ | $$ (\phi, z) \mapsto ((1 + z \sin (\phi/2) ) \cos \phi, (1 + z \sin (\phi/2) ) \sin \phi, z \cos(\phi/ | ||
+ | Question: is the induced metric on $M$ flat? | ||
+ | {{ : | ||
+ | |||
+ | 3. (Wiggly band): Let $M$ be a submanifold of $\R^3$ defined by the following embedding | ||
+ | $$ \R \times (-\delta, \delta) \to \R^3 $$ | ||
+ | $$ (t, z) \mapsto ( z \cos ( \sin t ) , z \sin (\sin t) ), t) $$ | ||
+ | Prove that the metric is not flat. For example, compute the curvature associated to the Levi-Cevita connection, and show that it is not identically zero. | ||
+ | {{ : | ||
+ | |||
+ | 4. Exercise 4.1.20 in [Ni]. Let $G$ be a Lie group. $X, Y \in T_e G$. Show that the parallel transport of $X$ along $exp(t Y)$ is given by | ||
+ | $$ L_{\exp( (t/2) Y )*} R_{\exp( (t/2) Y )*} X. $$ | ||
+ | |||
+ | 5. Compute the Killing form for $su(2)$ and $sl(2, \R)$. | ||