On a Theorem of Haupt and Wirtinger concerning the Periods of a Differential of the First Kind, and a Related Topological Theorem

On a closed Riemann surface Ra of genus g there exist g linearly independent differentials of the first kind, wu • • • , ws,.and their integrals around 2g canonical cycles or retrosections, Oi, • • • , a„, bi, ••• , ba, are usually put together to form a gX2g matrix (an; ßn) = (A; B), i, j =1, • • • , g, where a,-,is the integral of w{ over a,. 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~lB. The whole gX2g matrix is then of the form (7, Z) where Z = A~lB 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-\-iY, where X and F 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 gX2g matrix (A ; B) where A is nonsingular and A_1B 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~XB is symmetric, it has g(g + l)/2 independent complex entries, which may be taken as the coordinates of a point in complex g(g + l)/2-space, C('+1)l2. 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-