Sasakian Statistical Manifolds with Semi-Symmetric Metric Connection

Ahmet Kazan; Sema Kazan

doi:10.32323/ujma.439013

Research Article

Sasakian Statistical Manifolds with Semi-Symmetric Metric Connection

Year 2018, Volume: 1 Issue: 4, 226 - 232, 20.12.2018

Ahmet Kazan Sema Kazan

https://doi.org/10.32323/ujma.439013

Cited By: 8

Abstract

In the present paper, firstly we express the relation between the semi-symmetric metric connection $\tilde{\nabla}$ and the torsion-free connection $\nabla$ and obtain the relation between the curvature tensors $\tilde{R}$ of $\tilde{\nabla}$ and $R$ of $\nabla$. After, we obtain these relations for $\tilde{\nabla}$ and the dual connection $\nabla^{\ast}.$ Also, we give the relations between the curvature tensor $\tilde{R}$ of semi-symmetric metric connection $\tilde{\nabla}$ and the curvature tensors $R$ and $R^{\ast}$ of the connections $\nabla$ and $\nabla^{\ast}$ on Sasakian statistical manifolds, respectively. We obtain the relations between the Ricci tensor (and scalar curvature) of semi-symmetric metric connection $\tilde{\nabla}$ and the Ricci tensors (and scalar curvatures) of the connections $\nabla$ and $\nabla^{\ast}.$ Finally, we construct an example of a 3-dimensional Sasakian manifold with statistical structure admitting the semi-symmetric metric connection in order to verify our results.

Keywords

Sasakian Manifolds, Statistical Structure, Dual Connection, Semi-Symmetric Metric Connection, Statistical Structure

References

[1] C.R. Rao, Information and accuracy attainable in the estimation of statistical parameters, Bull. Calcutta Math. Soc., 37 (1945), 81–91.
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[3] A. S. Diallo, L. Todjihounde, Dualistic structures on twisted product manifolds, Global J. Adv. Res. Cl. Mod. Geom., 4(1) (2015), 35-43.
[4] S. Amari, Differential-geometrical methods in statistics, Lecture Notes in Statist., 28, Springer, New York, 1985.
[5] A. M. Blaga, M. Crasmareanu, Golden statistical structures, Comptes rendus de l’Acad emie bulgare des Sci., 69(9) (2016), 1113-1120.
[6] O. Calin, C. Udris¸te, Geometric modeling in probability and statistics, Springer, 2014.
[7] H. Furuhata, Hypersurfaces in statistical manifolds, Differential Geom. Appl., 27 (2009), 420-429.
[8] T. Kurose, Dual connections and affine geometry, Math. Z., 203 (1990), 115-121.
[9] H. Matsuzoe, J. I. Takeuchi, S. I. Amari, Equiaffine structures on statistical manifolds and Bayesian statistics, Differential Geom. Appl., 24 (2006), 567–578.
[10] M. Noguchi, Geometry of statistical manifolds, Differential Geom. Appl., 2 (1992), 197-222.
[11] S. Sasaki, On differentiable manifolds with certain structures which are closely related to almost contact structure, I, Tohoku Math. J., 12(2), (1960), 459–476.
[12] K. Kenmotsu, A class of almost contact Riemannian manifolds, Tohoku Math. J., 24 (1972), 93–103.
[13] J. A. Oubina, New classes of almost contact metric structures, Publ. Math. Debrecen, 32 (1985), 187-193.
[14] A. D. Vilcu, G. E. Vilcu, Statistical manifolds with almost quaternionic structures and quaternionic Kahler-like statistical submersions, Entropy, 17 (2015), 6213-6228.
[15] H. Furuhata, I. Hasegawa, Y. Okuyama, K. Sato, M. H. Shahid, Sasakian statistical manifolds, J. Geom. Phys., 117 (2017), 179-186.
[16] H. Furuhata, I. Hasegawa, Y. Okuyama, K. Sato, Kenmotsu statistical manifolds and warped product, J. Geom., (2017), doi: 10.1007/s00022-017-0403-1.
[17] J. Zhang, A note on curvature of a-connections of a statistical manifold, Ann. Inst. Statist. Math., 59 (2007), 161-170.
[18] H. A. Hayden, Subspace of a space with torsion, Proc. London Math. Soc. II Series, 34 (1932), 27–50.
[19] A. Friedmann, J. A. Schouten, U¨ ber die geometric der halbsymmetrischen U¨ bertragung, Math. Z., 21 (1924), 211–223.
[20] D. E. Blair, Contact manifolds in Riemannian geometry, Lect. Notes Math., 509, Springer, 1976.
[21] K. Yano, On semi-symmetric connection, Rev. Roumaine Math. Pures Appl., 15 (1970), 1570–1586.

Year 2018, Volume: 1 Issue: 4, 226 - 232, 20.12.2018

Ahmet Kazan Sema Kazan

https://doi.org/10.32323/ujma.439013

Cited By: 8

Abstract

References

[1] C.R. Rao, Information and accuracy attainable in the estimation of statistical parameters, Bull. Calcutta Math. Soc., 37 (1945), 81–91.
[2] N. Ay, W. Tuschmann, Dually flat manifolds and global information geometry, Open Syst. Inf. Dyn., 9 (2002), 195-200.
[3] A. S. Diallo, L. Todjihounde, Dualistic structures on twisted product manifolds, Global J. Adv. Res. Cl. Mod. Geom., 4(1) (2015), 35-43.
[4] S. Amari, Differential-geometrical methods in statistics, Lecture Notes in Statist., 28, Springer, New York, 1985.
[5] A. M. Blaga, M. Crasmareanu, Golden statistical structures, Comptes rendus de l’Acad emie bulgare des Sci., 69(9) (2016), 1113-1120.
[6] O. Calin, C. Udris¸te, Geometric modeling in probability and statistics, Springer, 2014.
[7] H. Furuhata, Hypersurfaces in statistical manifolds, Differential Geom. Appl., 27 (2009), 420-429.
[8] T. Kurose, Dual connections and affine geometry, Math. Z., 203 (1990), 115-121.
[9] H. Matsuzoe, J. I. Takeuchi, S. I. Amari, Equiaffine structures on statistical manifolds and Bayesian statistics, Differential Geom. Appl., 24 (2006), 567–578.
[10] M. Noguchi, Geometry of statistical manifolds, Differential Geom. Appl., 2 (1992), 197-222.
[11] S. Sasaki, On differentiable manifolds with certain structures which are closely related to almost contact structure, I, Tohoku Math. J., 12(2), (1960), 459–476.
[12] K. Kenmotsu, A class of almost contact Riemannian manifolds, Tohoku Math. J., 24 (1972), 93–103.
[13] J. A. Oubina, New classes of almost contact metric structures, Publ. Math. Debrecen, 32 (1985), 187-193.
[14] A. D. Vilcu, G. E. Vilcu, Statistical manifolds with almost quaternionic structures and quaternionic Kahler-like statistical submersions, Entropy, 17 (2015), 6213-6228.
[15] H. Furuhata, I. Hasegawa, Y. Okuyama, K. Sato, M. H. Shahid, Sasakian statistical manifolds, J. Geom. Phys., 117 (2017), 179-186.
[16] H. Furuhata, I. Hasegawa, Y. Okuyama, K. Sato, Kenmotsu statistical manifolds and warped product, J. Geom., (2017), doi: 10.1007/s00022-017-0403-1.
[17] J. Zhang, A note on curvature of a-connections of a statistical manifold, Ann. Inst. Statist. Math., 59 (2007), 161-170.
[18] H. A. Hayden, Subspace of a space with torsion, Proc. London Math. Soc. II Series, 34 (1932), 27–50.
[19] A. Friedmann, J. A. Schouten, U¨ ber die geometric der halbsymmetrischen U¨ bertragung, Math. Z., 21 (1924), 211–223.
[20] D. E. Blair, Contact manifolds in Riemannian geometry, Lect. Notes Math., 509, Springer, 1976.
[21] K. Yano, On semi-symmetric connection, Rev. Roumaine Math. Pures Appl., 15 (1970), 1570–1586.

There are 21 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Articles
Authors	Ahmet Kazan Sema Kazan
Publication Date	December 20, 2018
Submission Date	June 29, 2018
Acceptance Date	August 3, 2018
Published in Issue	Year 2018 Volume: 1 Issue: 4

Cite

APA	Kazan, A., & Kazan, S. (2018). Sasakian Statistical Manifolds with Semi-Symmetric Metric Connection. Universal Journal of Mathematics and Applications, 1(4), 226-232. https://doi.org/10.32323/ujma.439013
AMA	Kazan A, Kazan S. Sasakian Statistical Manifolds with Semi-Symmetric Metric Connection. Univ. J. Math. Appl. December 2018;1(4):226-232. doi:10.32323/ujma.439013
Chicago	Kazan, Ahmet, and Sema Kazan. “Sasakian Statistical Manifolds With Semi-Symmetric Metric Connection”. Universal Journal of Mathematics and Applications 1, no. 4 (December 2018): 226-32. https://doi.org/10.32323/ujma.439013.
EndNote	Kazan A, Kazan S (December 1, 2018) Sasakian Statistical Manifolds with Semi-Symmetric Metric Connection. Universal Journal of Mathematics and Applications 1 4 226–232.
IEEE	A. Kazan and S. Kazan, “Sasakian Statistical Manifolds with Semi-Symmetric Metric Connection”, Univ. J. Math. Appl., vol. 1, no. 4, pp. 226–232, 2018, doi: 10.32323/ujma.439013.
ISNAD	Kazan, Ahmet - Kazan, Sema. “Sasakian Statistical Manifolds With Semi-Symmetric Metric Connection”. Universal Journal of Mathematics and Applications 1/4 (December 2018), 226-232. https://doi.org/10.32323/ujma.439013.
JAMA	Kazan A, Kazan S. Sasakian Statistical Manifolds with Semi-Symmetric Metric Connection. Univ. J. Math. Appl. 2018;1:226–232.
MLA	Kazan, Ahmet and Sema Kazan. “Sasakian Statistical Manifolds With Semi-Symmetric Metric Connection”. Universal Journal of Mathematics and Applications, vol. 1, no. 4, 2018, pp. 226-32, doi:10.32323/ujma.439013.
Vancouver	Kazan A, Kazan S. Sasakian Statistical Manifolds with Semi-Symmetric Metric Connection. Univ. J. Math. Appl. 2018;1(4):226-32.