The Determinant Inner Product and the Heisenberg Product of Sym(2)

Abstract

The aim of this work is to introduce and study the nondegenerate inner product det inducedby the determinant map on the space Sym(2) of symmetric 2 2 real matrices. This symmetricbilinear form of index 2 defines a rational symmetric function on the pairs of rays in the planeand an associated function on the 2-torus can be expressed with the usual Hopf bundle projectionS3 ! S2( 12 ). Also, the product det is treated with complex numbers by using the Hopfinvariant map of Sym(2) and this complex approach yields a Heisenberg product on Sym(2).Moreover, the quadratic equation of critical points for a rational Morse function of height typegenerates a cosymplectic structure on Sym(2) with the unitary matrix as associated Reeb vector andwith the Reeb 1-form being half of the trace map.