Question

(a) A diffeomorphism : S1 S2 is area-preserving if the area of any region Rc S is equal to the area of 4(R). Show that if is area-preserving and conformal, then is a local isometry (b) Show that the Mercator's projection (defined in do Carmo, chapter 4-2, exercise 16) is not area ea-preserving (c) Lambert's cylindrical projection projects L = S2 \ {N,S}, the unit sphere minus the north pole N and the south pole S, into a the unit circle as cylinder on follows: х у лL:(х, у, 2). 2 x-+y° + y° Show that T preserves area, but is neither a local isometry conformal nor Mercator (left) and Lambert (right) projections are used in cartography (source: Wikipedia).

Answer 1

SOLUTION

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