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

Year 2021, Volume: 14 Issue: 1, 145 - 156, 15.04.2021

Mircea Crasmareanu

https://doi.org/10.36890/iejg.754557

Cited By: 2

Abstract

The aim of this work is to introduce and study the nondegenerate inner product $<\cdot , \cdot >_{det}$ induced by the determinant map on the space $Sym(2)$ of symmetric $2\times 2$ 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 $S^3\rightarrow S^2(\frac{1}{2})$. Also, the product $<\cdot , \cdot >_{det}$ 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.

Keywords

Symmetric matrix, determinant, Hopf bundle, Hopf invariant

References

• [1] Bejancu, A.: Groupes de Lie-Banach et espaces localement symétriques. An. Stiint. Univ. Al. I. Cuza Iasi, Sect. Ia. 18, 401–405 (1972).
• [2] Bejancu, A.: Sur l’existence d’une structure de groupe de Lie-Banach local. C. R. Acad. Sci., Paris, Sér. A. 276, 61–64 (1973).
• [3] Bos, L., Slawinski, M. A.: Proof of validity of first-order seismic traveltime estimates. GEM. Int. J. Geomath. 2(2), 255-263 (2011).
• [4] Crasmareanu, M.: Adapted metrics and Webster curvature on three classes of 3-dimensional geometries. Int. Electron. J. Geom. 7(2), 37–46 (2014).
• [5] Crasmareanu, M.: A complex approach to the gradient-type deformation of conics. Bull. Transilv. Univ. Bra¸sov, Ser. III, Math. Inform. Phys. 10(59), 59–62 (2017).
• [6] Crasmareanu, M.: Conics from symmetric Pythagorean triple preserving matrices. Int. Electron. J. Geom. 12(1), 85–92 (2019).
• [7] Crasmareanu, M.: Clifford product of cycles in EPH geometries and EPH-square of elliptic curves. An. Stiint. Univ. Al. I. Cuza Iasi Mat. 66(1), 147–160 (2020).
• [8] Crasmareanu, M.: Magic conics, their integer points and complementary ellipses. An. Stiint. Univ. Al. I. Cuza Iasi Mat. in press.
• [9] Crasmareanu, M., Plugariu, A.: New aspects on square roots of a real 2 x 2 matrix and their geometric applications. Math. Sci. Appl. E-Notes 6(1), 37–42 (2018).
• [10] González, M. O.: Classical complex analysis. Monographs and Textbooks in Pure and Applied Mathematics, 151. Marcel Dekker, Inc., New York, 1992.
• [11] Jensen, G. R., Musso, E., Nicolodi L.: Surfaces in classical geometries. A treatment by moving frames. Universitext. Springer, Cham, 2016.
• [12] Lee, J. M.: Introduction to Riemannian manifolds. Second edition. Graduate Texts in Mathematics, 176. Springer, Cham, 2018.
• [13] Nicolaescu, L. I.: Morse theory on Grassmannians. An. ¸Stiin¸t. Univ. Al. I. Cuza Ia¸si, Mat. 40(1), 25–46 (1994).
• [14] Nomizu, K., Pinkall, U.: Lorentzian geometry for 2 x 2 real matrices. Linear Multilinear Algebra 28(4), 207–212 (1990).
• [15] Özdemir, F., Crasmareanu, M.: Geometrical objects associated to a substructure. Turk. J. Math. 35(4), 717–728 (2011).
• [16] Rosenberg, S.: The Laplacian on a Riemannian manifold. An introduction to analysis on manifolds. London Mathematical Society Student Texts, 31. Cambridge University Press, Cambridge, 1997.
• [17] Wilkinson, A.: What are Lyapunov exponents, and why are they interesting? Bull. Am. Math. Soc. New Ser. 54(1) 79–105 (2017).