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Can a Sphere be Flattened?
Posted:
Nov 12, 2011 7:34 AM
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We consider 2D metric:
ds^2=d(theta)^2+Sin^2(theta)d(phi)^2 ------------- (1)
Is it possible to transform the above metric to the form:
ds^2=dx^2+dy^2 --------------------- (2)
Let¹s check:
Initially we write equation (2) in the form:
ds¹^2=dx^2+dy^2 ---(3)
For relations (1) and (3) we use the following transformations:
theta=f1(p,q)
phi=f2(p,q)
x=f3(p,q)
y=f4(p,q)
d(theta)=[del_f1/del_p ]*dp + [del_f1/del_q]* dq
d(phi)=[del_f2/del_p ]*dp + [del_f2/del_q]* dq
dx=[del_f3/del_p ]*dp + [del_f3/del_q]* dq
dy=[del_f4/del_p ]*dp + [del_f4/del_q]* dq
Using the above transformations in(1) and (3) we have:
ds^2=[(del_f1/del_p)^2+sin^2(f1)(del_f2/del_p)^2]*dp^2+[(del_f1/
del_q)^2+sin^2(f1)(del_f2/del_q)^2]dq^2+2[del_f1/del_p del_f1/del_q+
Sin^2(f1) del_f2/del_p del_f2/del_q]dp*dq ------------- (4)
And
ds¹^2=[(del_f3/del_p)^2+(del_f4/del_p)^2]*dp^2+[(del_f3/del_q)^2+
(del_f4/del_q)^2]dq^2+2[del_f3/del_p del_f3/del_q+ del_f4/del_p
del_f4/del_q]dp*dq ------------- (3)
To make ds^2=ds¹^2 we may consider the following equations:
[denoted by SET A]:
(del_f1/del_p)^2+sin^2(f1)(del_f2/del_p)^2= (del_f3/del_p)^2+(del_f4/
del_p)^2 ---------- (A1)
(del_f1/del_q)^2+sin^2(f1)(del_f2/del_q)^2= (del_f3/del_q)^2+(del_f4/
del_q)^2 --------- (A2)
del_f1/del_p del_f1/del_q+ Sin^2(f1) del_f2/del_p del_f2/del_q =
del_f3/del_p del_f3/del_q+ del_f4/del_p del_f4/del_q
------------- (A3)
If SET A has solutions for the functions f1,f2,f3 and f4 we are
passing from relation (1) to relation (2) by coordinate
transformation[ds^2 is preserved and it is being carried from one
manifold to another ]
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