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--- math214:hw11 [2020/04/10 13:26]
pzhou
+++ math214:hw11 [2020/04/15 10:47] (current)
pzhou
@@ Line 11: / Line 11: @@
 $$ (0, 2\pi] \times (-\delta, \delta) \to \R^3$$
 where
-$$ (\phi, z) \mapsto ((1 + z \sin (\phi/2) ) \cos \phi, (1 + z \cos (\phi/2) ) \sin \phi, z \cos(\phi/2) ) $$
+$$ (\phi, z) \mapsto ((1 + z \sin (\phi/2) ) \cos \phi, (1 + z \sin (\phi/2) ) \sin \phi, z \cos(\phi/2) ) $$
 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, i.e, the curvature associated to the Levi-Cevita connection is non-zero.
+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