Lecture Notes

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====== Homework 12 ======



1. (Geodesics are length extremizing). Recall the following facts
   * Consider the space $\R^2$. If we equip $\R^2$ with the flat metric, then it is well-known that every of two points has a unique geodesic connecting them. 
   * Consider the sphere $S^2$, if we equip $S^2$ with the usual round metric, then every pair of two points (not antipodal pairs) has exactly two geodesics connecting them. 
Here comes the question: we will equip $\R^2$ with a metric as following
$$ g = \iota^* (g_{\R^3}), \quad \iota: \R^2 \to \R^3, \quad (x,y) \mapsto (x,y, h_r(x,y))$$
where 
$$h_r(x,y) = r^{-2} e^{ - (x^2+y^2)/r^2}$$
is a Guassian peak with radius $r$ and height $r^{-2}$. Answer the following question without doing computation: 
   - Let $p = (-1,0)$, $q = (1,0)$. As $r \to \infty$, how many geodesics are there between $p$ and $q$? 
   - As $r \to 0$, how does the amount of geodesics between $p$ and $q$ changes? 
   - Is it possible that for any finite $r$, there are only finitely many geodesics between $p$ and $q$? 

Here is a picture, and the {{ :math214:geodesics.nb.zip |Mathematica program}} to make that picture (FYI, you can [[https://software.berkeley.edu/mathematica | use Mathematica for free]] as Berkeley student!) In the program, I fixed the initial point, and varies shooting angle, and the peak height. 
{{ :math214:bending.png |}} 

And here is a video: {{ :math214:zoom_0.mp4 | varying the bumps and angles. }}

2. Let $S$ be the submanifold of $\R^3$, that arises as the graph of $x^2 - y^2$. Compute the second fundamental form of $S$ at $x=0, y=0$. 

3. Let $G$ be a compact Lie group with a bi-invariant metric $\la -,- \ra$. Let $X, Y, Z$ be left-invariant vector fields. (try to do it yourself before checking Example 4.2.11 in [Ni]). Show that
$$ R(X, Y) Z = (-1/4) [ [X, Y], Z] $$

4. (Cartan 3-form). Same setup as 3. There is $3$-form $B$ on $G$, satisfying
$$B(X, Y, Z) = \la [X, Y], Z \ra. $$
Show that this form is closed. In the case $G = SU(2) \cong S^3$, can you recognize this $3$-form as something familiar? 

Hint: use the [[https://en.wikipedia.org/wiki/Exterior_derivative#In_terms_of_invariant_formula | invariant formula for exterior derivative]], and plug in the left-invariant vector fields. 

5. (Normal Coordinate.) Ex 4.1.43. Hint: choose nice coordinate and nice basis.
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