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Torus, two graphs and their equivalency

  • Thread starter parsifal
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  • #1
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Homework Statement


Two graphs defined for a two dimensional torus:
[tex]f_1(t) = (\frac{1}{\sqrt{2}}(a\ +\ b\ sin\ t),\frac{1}{\sqrt{2}}(a\ +\ b\ sin\ t),b\ cos\ t),\
t \in (-\frac{\pi}{2},\frac{\pi}{2})[/tex]
[tex]f_2(t) = (a\ cos(t+\frac{\pi}{4}),a\ sin(t+\frac{\pi}{4}),b),\
t \in (-\frac{\pi}{4},\frac{\pi}{4})[/tex]

Then [tex]f_1(0)=f_2(0)=(\frac{a}{\sqrt{2}},\frac{a}{\sqrt{2}},b)[/tex]

Are the graphs equivalent at a point [tex](\frac{a}{\sqrt{2}},\frac{a}{\sqrt{2}},b)[/tex]?

Homework Equations


Graphs f1 and f2 on manifold M are equivalent at a point m (m is in an open group U), if for some chart fc(q1,...,qn): U -> Rn of manifold M
[tex]\frac{d}{dt}q^i(f_1(t))=\frac{d}{dt}q^i(f_2(t))\ |_{t=0}[/tex]

I also know that q is a coordinate function:
[tex]q^i:=pr^i \circ f_c[/tex] where the projection [tex]pr^i:R^n \rightarrow R, (x^1,...,x^n) \mapsto x^i[/tex]

I'm also told that if [tex]q^i:=pr^i \circ f_c[/tex] then f can be written as: [tex]f_c=(q^1,...,q^n)[/tex]

The Attempt at a Solution


I can't actually give anything of my own to go with this. Last week I posted a question relating to this problem, but as anyone hasn't been able to comprehend my output, I'm hoping that someone could help me with the problem directly.

So the problem is I really don't understand the behaviour of the coordinate function in this, but I'll happily read any kind of suggestions (using the method mentioned in 2. or not) for solving the problem.
 
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Answers and Replies

  • #2
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No answers. It's not my intention to be impatient, however. Instead, I was wondering if this whole "equivalency of graphs" is not as widely known as I though. My friend majoring in math has never heard of it, and it was actually presented to me on a physics course (not that this is a proof of anything, but got me thinking).

I tried to search the internet using the relevant words as keywords, but could not find anything where this equivalency were the substantial part of the document. So possibly I don't know the correct terms to search with (as I'm very unfamiliar with English math vocabulary). Could anybody give me any hints on which kind of math material such a topic could be covered?
 
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