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--- math214:hw11 [2020/04/10 13:08]
pzhou created
+++ math214:hw11 [2020/04/15 10:47] (current)
pzhou
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-. 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
+. 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
-$$ (0, 2\pi] \times (-\delta, \delta) \mapsto \R^3, (\phi, z) \mapsto ((1 + z \sin (\phi/2)) \cos \phi, (1 + z \cos (\phi/2)) \sin \phi, z \cos(\phi/2)) $$
+$$ (0, 2\pi] \times (-\delta, \delta) \to \R^3$$
+where
+$$ (\phi, z) \mapsto ((1 + z \sin (\phi/2) ) \cos \phi, (1 + z \sin (\phi/2) ) \sin \phi, z \cos(\phi/2) ) $$
+Question: is the induced metric on $M$ flat?
+{{ :math214:mobius.png |}}
+. (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.
+{{ :math214:wiggly.png?600 |}}
+. 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
+$$ L_{\exp( (t/2) Y )*}  R_{\exp( (t/2) Y )*} X. $$
+. Compute the Killing form for $su(2)$ and $sl(2, \R)$.

Lecture Notes