On a theorem of Haupt and Wirtinger concerning the periods of a differential of the first kind, and a related topological theorem

Abstract

Riemann showed that the entries in this matrix were far from arbitrary. In fact, both of the square matrices A and B are nonsingular, making it possible to choose a new basis for the-set of differentials of the first kind in such a way that A is replaced by the gXg identity matrix and B is replaced by A-'B. The whole gX2g matrix is then of the form (I, Z) where Z =A-'B is a square matrix with complex entries. If each element of Z is written as the sum of its real and imaginary parts, then Z itself may be written as a sum of two matrices, Z = X+i Y, where X and Y have real entries. Then the Riemann relations between the entries in (A; B) are equivalent to the statement that Z is now symmetric (from which it follows that both X and Y are symmetric) and that Y is in fact positive definite. An arbitrary g X 2g matrix (A; B) where A is nonsingular and A-'B is symmetric and has positive definite imaginary part is often called (after Scorza) a Riemann matrix, but it is not true that every Riemann matrix, in this sense, arises as the Riemann matrix associated with a Riemann surface. Let (A; B) be an arbitrary Riemann matrix.2 Since Z=A-'B is symmetric, it has g(g+1)/2 independent complex entries, which may be taken as the coordinates of a point in complex g(g+1)/2-space, Cg(+1)12. The set of points arising thus from the totality of all Riemann matrices of genus g is an open set, being restricted only by the condition that the imaginary part of Z be positive definite, and therefore has the same dimension as the space. But it is known that if the set of all Riemann surfaces of genus g is parametrized in any reasonable fashion, then for g= 1, one complex param