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math214:hw11 [2020/04/10 13:43] pzhou |
math214:hw11 [2020/04/15 10:47] (current) pzhou |
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$$ (0, 2\pi] \times (-\delta, \delta) \to \R^3$$ | $$ (0, 2\pi] \times (-\delta, \delta) \to \R^3$$ | ||
where | where | ||
- | $$ (\phi, z) \mapsto ((1 + z \sin (\phi/2) ) \cos \phi, (1 + z \cos (\phi/2) ) \sin \phi, z \cos(\phi/ | + | $$ (\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? | Question: is the induced metric on $M$ flat? | ||
+ | |||
+ | {{ : | ||
3. (Wiggly band): Let $M$ be a submanifold of $\R^3$ defined by the following embedding | 3. (Wiggly band): Let $M$ be a submanifold of $\R^3$ defined by the following embedding | ||
$$ \R \times (-\delta, \delta) \to \R^3 $$ | $$ \R \times (-\delta, \delta) \to \R^3 $$ | ||
$$ (t, z) \mapsto ( z \cos ( \sin t ) , z \sin (\sin t) ), t) $$ | $$ (t, z) \mapsto ( z \cos ( \sin t ) , z \sin (\sin t) ), t) $$ | ||
- | Prove that the metric is not flat, i.e, the curvature associated to the Levi-Cevita connection is non-zero. If you are brave enough, find the curvature. | + | Prove that the metric is not flat. For example, compute |
{{ : | {{ : | ||