Equiareal map
In differential geometry, an equiareal map, sometimes called an authalic map, is a smooth map from one surface to another that preserves the areas of figures. == Properties == If M and N are two Riemannian (or pseudo-Riemannian) surfaces, then an equiareal map f from M to N can be characterized by any of the following equivalent conditions: The surface area of f(U) is equal to the area of U for every open set U on M. The pullback of the area element μN on N is equal to μM, the area element on M. At each point p of M, and tangent vectors v and w to M at p, | d f p ( v ) ∧ d f p ( w ) | = | v ∧ w | {\displaystyle {\bigl |}df_{p}(v)\wedge df_{p}(w){\bigr |}=|v\wedge w|\,} where ∧ {\textstyle \wedge } denotes the Euclidean wedge product of vectors and df denotes the pushforward along f.