Can a Sphere be Flattened?

Discussion in 'Math Research' started by Anamitra Palit, Nov 12, 2011.

  1. 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 ]
     
    Anamitra Palit, Nov 12, 2011
    #1
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