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--- math214:hw12 [2020/04/26 08:20]
pzhou
+++ math214:hw12 [2026/02/21 14:41] (current)
@@ Line 15: / Line 15: @@
    - 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!)
+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. }}
 . 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 $x=0, y=0$.
@@ Line 26: / Line 28: @@
 $$B(X, Y, Z) = \la [X, Y], Z \ra. $$
 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.
 . (Normal Coordinate.) Ex 4.1.43. Hint: choose nice coordinate and nice basis.

Lecture Notes