Araştırma Makalesi
BibTex RIS Kaynak Göster
Yıl 2021, Cilt: 14 Sayı: 1, 145 - 156, 15.04.2021
https://doi.org/10.36890/iejg.754557

Öz

Kaynakça

  • [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).

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

Yıl 2021, Cilt: 14 Sayı: 1, 145 - 156, 15.04.2021
https://doi.org/10.36890/iejg.754557

Öz

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.

Kaynakça

  • [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).
Toplam 17 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Araştırma Makalesi
Yazarlar

Mircea Crasmareanu 0000-0002-5230-2751

Yayımlanma Tarihi 15 Nisan 2021
Kabul Tarihi 3 Mart 2021
Yayımlandığı Sayı Yıl 2021 Cilt: 14 Sayı: 1

Kaynak Göster

APA Crasmareanu, M. (2021). The Determinant Inner Product and the Heisenberg Product of $Sym(2)$. International Electronic Journal of Geometry, 14(1), 145-156. https://doi.org/10.36890/iejg.754557
AMA Crasmareanu M. The Determinant Inner Product and the Heisenberg Product of $Sym(2)$. Int. Electron. J. Geom. Nisan 2021;14(1):145-156. doi:10.36890/iejg.754557
Chicago Crasmareanu, Mircea. “The Determinant Inner Product and the Heisenberg Product of $Sym(2)$”. International Electronic Journal of Geometry 14, sy. 1 (Nisan 2021): 145-56. https://doi.org/10.36890/iejg.754557.
EndNote Crasmareanu M (01 Nisan 2021) The Determinant Inner Product and the Heisenberg Product of $Sym(2)$. International Electronic Journal of Geometry 14 1 145–156.
IEEE M. Crasmareanu, “The Determinant Inner Product and the Heisenberg Product of $Sym(2)$”, Int. Electron. J. Geom., c. 14, sy. 1, ss. 145–156, 2021, doi: 10.36890/iejg.754557.
ISNAD Crasmareanu, Mircea. “The Determinant Inner Product and the Heisenberg Product of $Sym(2)$”. International Electronic Journal of Geometry 14/1 (Nisan 2021), 145-156. https://doi.org/10.36890/iejg.754557.
JAMA Crasmareanu M. The Determinant Inner Product and the Heisenberg Product of $Sym(2)$. Int. Electron. J. Geom. 2021;14:145–156.
MLA Crasmareanu, Mircea. “The Determinant Inner Product and the Heisenberg Product of $Sym(2)$”. International Electronic Journal of Geometry, c. 14, sy. 1, 2021, ss. 145-56, doi:10.36890/iejg.754557.
Vancouver Crasmareanu M. The Determinant Inner Product and the Heisenberg Product of $Sym(2)$. Int. Electron. J. Geom. 2021;14(1):145-56.