A finite-difference method on a Riemann surface
Hisao Mizumoto · Hiroshima Mathematical Journal · 1973
1. Polyangulation.Let E 2 be the euclidean plane.By a euclidean 0-simplex we mean a point on E 2 .By a euclidean 1-simplex we mean a closed line segment or a closed circular arc.By a euclidean 2-simplex we mean a closed polygon surrounded by a finite number (^2) of segments and circular arcs.A lune (biangle) and a triangle are also admitted as a euclidean 2-simplex.Let F be a 2-dimensional orientable manifold.By 0-simplex q 9 1-simplex a and 2-simplex M on F we mean a pair of euclidean 0-simplex q e , 1-simplex a e and 2-simplex M e respectively, and one-to-one bicontinuous mappings φ of q e , a e and M e respectively into F. We shall write q = [q e , ], a=[a e , φ~\ and M = [M e , ].The images of q e 9 a e and M e under φ are called the carriers of q, a and M respectively, and are denoted by \q\, \a\ and \M\ respectively; that is, φ(q e ) = \q\, φ(a e ) = \a\ and φ(M e ) = \M\.M is called a polygon on F, and the images of the edges and vertices of M e are called edges and vertices of M. Each edge of M is a 1-simplex and each vertex of M is a 0-simρlex, We say that a point p