Two-dimensional geometries with elementary areas

Herbert Busemann
1947 Bulletin of the American Mathematical Society  
Whereas area in spaces with a smooth Riemann metric has been widely studied, very little is known regarding area in spaces with general metrics. It is natural to ask, first, in which general spaces the most familiar types of formulas for area hold. The present note answers this question in two cases for two-dimensional spaces in which the geodesic connection is locally unique. 1 It shows under very weak assumptions regarding the nature of area: I. If {and only if) locally an area exists for
more » ... area exists for which triangles with equal sides have equal area, then the space is a locally isometric map of either the euclidean plane, or a hyperbolic plane, or a sphere. Consequently, Hero's and the corresponding non-euclidean formulas 2 are (up to constant factors) the only possible formulas for area in terms of the sides, and each formula is characteristic for its respective geometry. II. If {and only if) locally an area a exists such that the area of the triangle pab depends only on p, the local branch of the geodesic Q that contains the segment $(a, b), and the distance ab (the euclidean geometry is, of course, the special case a = p$-ab/2), then the space is a locally isometric map of a Minkowski plane. The exact hypotheses regarding the space R are these: (1) R is a metric space. (2) R is finitely compact. (3) R is two-dimensional. (4) R is convex. If xy denotes the distance of x and y, let {xyz) denote the statement that the three points x, y, z are different and that xy+yz -xz. (5) Every point p has a neighborhood U{p) such that for any two different points x, y in U{p) a point z with {xyz) exists. (6) If {xyzi), {xyz 2 ) and yz\ -yz 2 , then z x =z 2 . z The following facts are known to hold in R: If S{p, p) denotes the set of points x with px0 exists such that: S{p, Sp{p)) is homeomorphic to a circular disk [l, p. 29]. The segment %{a, b)
doi:10.1090/s0002-9904-1947-08814-5 fatcat:5xhj4blkfvbxdcw72su7cdxx6q