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 $<\cdot , \cdot >_{det}$ induced by determinant map on space $Sym(2)$ symmetric $2\times 2$ real matrices. This bilinear form index $2$ defines a rational function pairs rays in plane an associated $2$-torus can be expressed with usual Hopf bundle projection $S^3\rightarrow S^2(\frac{1}{2})$. Also, treated complex numbers using invariant approach yields Heisenberg $Sym(2)$. Moreover, quadratic equation critical points for Morse height type generates cosymplectic structure unitary matrix as Reeb vector $1$-form being half trace map.