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

The aim of this work is to introduce and study the nondegenerate inner product < .,. >(det) induced by the determinant map on the space Sym(2) of symmetric 2 x 2 real matrices. This symmetric bilinear form of index 2 defines a rational symmetric function on the pairs of rays in the plane and an associated function on the 2-torus can be expressed with the usual Hopf bundle projection S-3 -> S-2 (1/2). Also, the product < .,. >(det) is treated with complex numbers by using the Hopf invariant 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 type generates a cosymplectic structure on Sym(2) with the unitary matrix as associated Reeb vector and with the Reeb 1-form being half of the trace map.