Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

CURVATURES OF SEMI-SYMMETRIC METRIC CONNECTIONS ON STATISTICAL MANIFOLDS

  • Received : 2020.01.02
  • Accepted : 2020.08.28
  • Published : 2021.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

By using a statistical connection, we define a semi-symmetric metric connection on statistical manifolds and study the geometry of these manifolds and their submanifolds. We show the symmetry properties of the curvature tensor with respect to the semi-symmetric metric connections. Also, we prove the induced connection on a submanifold with respect to a semi-symmetric metric connection is a semi-symmetric metric connection and the second fundamental form coincides with the second fundamental form of the Levi-Civita connection. Furthermore, we obtain the Gauss, Codazzi and Ricci equations with respect to the new connection. Finally, we construct non-trivial examples of statistical manifolds admitting a semi-symmetric metric connection.

Keywords

References

  1. N. S. Agashe and M. R. Chafle, A semi-symmetric nonmetric connection on a Riemannian manifold, Indian J. Pure Appl. Math. 23 (1992), no. 6, 399-409.
  2. S. Amari and H. Nagaoka, Methods of information geometry, translated from the 1993 Japanese original by Daishi Harada, Translations of Mathematical Monographs, 191, American Mathematical Society, Providence, RI, 2000. https://doi.org/10.1090/mmono/191
  3. K. A. Arwini and C. T. J. Dodson, Information geometry, Lecture Notes in Mathematics, 1953, Springer-Verlag, Berlin, 2008. https://doi.org/10.1007/978-3-540-69393-2
  4. M. E. Aydin, A. Mihai, and I. Mihai, Generalized Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature, Bull. Math. Sci. 7 (2017), no. 1, 155-166. https://doi.org/10.1007/s13373-016-0086-1
  5. O. Calin and C. Udriste, Geometric Modeling in Probability and Statistics, Springer, Cham, 2014. https://doi.org/10.1007/978-3-319-07779-6
  6. U. C. De and A. Barman, On a type of semisymmetric metric connection on a Riemannian manifold, Publ. Inst. Math. (Beograd) (N.S.) 98(112) (2015), 211-218. https://doi.org/10.2298/PIM150317025D
  7. A. Friedmann and J. A. Schouten, Uber die Geometrie der halbsymmetrischen Uber-tragungen, Math. Z. 21 (1924), no. 1, 211-223. https://doi.org/10.1007/BF01187468
  8. A. Haseeb and R. Prasad, Certain curvature conditions in Kenmotsu manifolds with respect to the semi-symmetric metric connection, Commun. Korean Math. Soc. 32 (2017), no. 4, 1033-1045. https://doi.org/10.4134/CKMS.c160266
  9. S. Kazan and A. Kazan, Sasakian statistical manifolds with semi-symmetric metric connection, Univers. J. Math. Appl. 1 (2018), 226-232. https://doi.org/10.32323/ujma.439013
  10. C. Murathan and C. Ozgur, Riemannian manifolds with a semi-symmetric metric connection satisfying some semisymmetry conditions, Proc. Est. Acad. Sci. 57 (2008), no. 4, 210-216. https://doi.org/10.3176/proc.2008.4.02
  11. R. N. Singh, S. K. Pandey, G. Pandey, and K. Tiwari, On a semi-symmetric metric connection in an (ε)-Kenmotsu manifold, Commun. Korean Math. Soc. 29 (2014), no. 2, 331-343. https://doi.org/10.4134/CKMS.2014.29.2.331
  12. K. Takano, Statistical manifolds with almost contact structures and its statistical submersions, J. Geom. 85 (2006), no. 1-2, 171-187. https://doi.org/10.1007/s00022-006-0052-2
  13. K. Yano, On semi-symmetric metric connection, Rev. Roumaine Math. Pures Appl. 15 (1970), 1579-1586.