Adam Haunch (University of Manchester)
A.Haunch@postgrad.manchester.ac.uk We consider rational functions with coefficients in the Grassmann algebra. Such functions naturally arise in connection with surfaces in superspace, i.e., the space which possesses both commuting and anticommuting coordinates.
We consider hypersurfaces in Euclidean superspace of dimension m|2n and introduce shape function of a surface as
R(z,S)= Ber (1+zS)
where S is the shape operator at a given point and z is a formal parameter. Here Ber is the analogue of determinant in supercase, which is a rational function of matrix entries rather than a polynomial. The coefficients of the expansion of the shape function over z are differential-geometric invariants of a surface. The shape function should be a ratio of polynomials of degrees m-1 and 2n respectively. However, a straightforward calculation can lead to polynomials of unexpectedly higher degrees containing common factors. Elimination of these factors can be a problem because of the presence of zero divisors. We develop a mechanism of such an elimination. We study properties of the shape function. In particular we discuss analogues of mean and Gaussian curvatures in supercase and suggest a generalisation of the tube formula.