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math214:hw12

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math214:hw12 [2020/04/20 02:30]
pzhou 		math214:hw12 [2026/02/21 14:41] (current)
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 where  where 
 $$h_r(x,y) = r^{-2} e^{ - (x^2+y^2)/r^2}$$ $$h_r(x,y) = r^{-2} e^{ - (x^2+y^2)/r^2}$$
-is a Guassian peak with radius $r$ and height $r^2$. Answer the following question without doing computation: +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 $q$?     - 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?     - 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 $q$?     - 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 $x=0, y=0$.  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 $x=0, y=0$. 
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 $$B(X, Y, Z) = \la [X, Y], Z \ra. $$ $$B(X, Y, Z) = \la [X, Y], Z \ra. $$
 Show that this form is closed. In the case $G = SU(2) \cong S^3$, can you recognize this $3$-form as something familiar?  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. 5. (Normal Coordinate.) Ex 4.1.43. Hint: choose nice coordinate and nice basis.
math214/hw12.1587349833.txt.gz · Last modified: 2026/02/21 14:44 (external edit)

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