Problem in my differential geometry final

In summary, the conversation discussed a problem involving a conformal equi-areal map and the use of the characterizations of "conformal" and "equiareal" to solve it. The equation E1G1-F1²=L²(E1G1-F1²) was derived and it was found that L=+1 or -1. However, for the map to be an isometry, L must be equal to 1. The discussion also touched on the definitions and signs of Ei and Gi. The question of how to reject L=-1 was also raised.
  • #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?
 
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  • #2
What are the definitions of the Ei and Gi? In particular, what can you say about their signs?
 
  • #3
Aww! :wink:
 

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