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Problem in my differential geometry final

  1. Dec 18, 2006 #1

    quasar987

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    There was a little problem in my final exam that went "Show that a conformal equi-areal map is an isometry".

    I invoqued the caracterisation of "conformal" that the two metrics are proportional, say by a proportionality function L: E1=LE2, F1=LF2, G1=LG2.

    Then I invoked the caracterisation of "equiareal" that the determinants of the two metrics must be equal: E1G1-F1²=E2G2-F2².

    Combining these two proporties yields the equation

    E1G1-F1²=L²(E1G1-F1²)

    which means that L²=1, which means that L=+1 or -1.

    But for the map to be an isometry, it must be L=1 (the two metrics must be equal), so how do we reject L=-1?
     
    Last edited: Dec 18, 2006
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  3. Dec 18, 2006 #2

    AKG

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    What are the definitions of the Ei and Gi? In particular, what can you say about their signs?
     
  4. Dec 18, 2006 #3

    quasar987

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    Aww! :wink:
     
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