Research Article
BibTex RIS Cite

Year 2019, Volume: 12 Issue: 1, 43 - 56, 27.03.2019

Ruken Görünüş İrem Küpeli Erken , Aziz Yazla Cengizhan Murathan

Abstract

References

  • [1] Amari, S., Differential-Geometrical Methods in Statistics. Lecture Notes in Statistics. 28 Springer, Berlin 1985.
  • [2] Aydın, M. E., Mihai, A. and Mihai, I., Some inequalities on submanifolds in statistical manifolds of constant curvature. Filomat 29(3) (2015), 465-477.
  • [3] Aydın, M.E., Mihai, A. and Mihai I., Generalized Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature. Bull. Math. Sci. 7(1) (2017), 155-166.
  • [4] Aydın, M.E. and Mihai I., Wintgen inequality for statistical surfaces. Math. Inequal. Appl. 22(1)(2019), 123–132.
  • [5] Aytimur, H. and Özgür, C., Inequalities for submanifolds in statistical manifolds of quasi-constant curvature. Ann. Polon. Math. 121 (2018), no. 3, 197–215.
  • [6] Blair D. E., Riemannian Geometry of Contact and Symplectic Manifolds. Boston. Birkhâuser 2002.
  • [7] Boyom, M. N., Aquib, M., Shahid M.H. and Jamali,M., Generalized Wintgen Type Inequality for Lagrangian Submanifolds in Holomorphic Statistical Space Forms. Frank Nielsen  Frédéric Barbaresco (Eds.) Geometric Science of Information Third International Conference, GSI 2017 Paris, France, November 7–9, 2017.
  • [8] Carriazo, A. and Perez-Garcia, M.J., Slant submanifolds in neutral almost contact pseudo-metric manifolds. Differ. Geom. Appl. 54 (2017), 71–80.
  • [9] Chen, B. Y., Mean curvature and shape operator of isometric immersions in real-space forms. Glasgow Math. J. 38 (1996), 87-97.
  • [10] Chen, Q. and Cui, Q., Normal scalar curvature and a pinching theorem in Sm  R and Hm  R. Science China Math. 54(9) (2011), 1977- 1984.
  • [11] De Smet, P. J., Dillen, F., Verstraelen, L. and Vrancken, L., A pointwise inequality in submanifold theory. Arch. Math. (Brno) 35 (1999), 115-128.
  • [12] Dillen, F., Fastenakels, J. and Van der Veken, J., Remarks on an inequality involving the normal scalar curvature. Pure and Applied Differential Geometry-PADGE 2007, 83-92, Ber. Math., Shaker Verlag, Aachen, 2007.
  • [13] Furuhata, H., Hypersurfaces in statistical manifolds. Diff. Geom. Appl. 27 (2009), 420-429.
  • [14] Furuhata, H., Hasegawa, I., Okuyama, Y. and Sato, K., Kenmotsu statistical manifolds and warped product. J. Geom. 108 (2017), 1175–1191.
  • [15] Ge, J. and Tang, Z., A proof of the DDVV conjecture and its equality case. Pacific J. Math. 237 (2008), 87-95.
  • [16] Kenmotsu, K., A class of contact Riemannian manifold. Tohoku Math. Journal 24 (1972), 93-103.
  • [17] Lauritzen, S., Statistical manifolds. In: Amari, S., Barndorff-Nielsen, O., Kass, R., Lauritzen, S., Rao, C.R. (eds.) Differential Geometry in Statistical Inference, 10, 163–216. IMS Lecture NotesInstitute of Mathematical Statistics, Hayward 1987.
  • [18] Lawn, M. and Ortega, M., A fundamental theorem for hypersurfaces in semi-Riemannian warped products. J. Geom. Phys. 90 (2015), 55-70. [19] Lu, Z., Normal scalar curvature conjecture and its applications. J. Funct. Analysis 261 (2011), 1284-1308.
  • [20] Mihai, I., On the generalizedWintgen inequality for Lagrangian submanifolds in complex space forms. Nonlinear Anal. 95 (2014), 714-720.
  • [21] Mihai, I., On the generalized Wintgen inequality for Legendrian submanifolds in Sasakian space forms. Tohoku Math. J. 69 (2017), 43-53.
  • [22] Murathan, C. and ¸Sahin, B., A study of Wintgen like inequality for submanifolds in statistical warped product manifolds. J. Geom. 109 (2018), 18 pp.
  • [23] Opozda, B., A sectional curvature for statistical structures. Linear Algebra Appl. 497 (2016), 134–161.
  • [24] Roth, J., A DDVV Inequality for submanifolds of warped products. Bull. Aust. Math. Soc. 95 (2017), 495–499.
  • [25] Simon, U., Affine Differential Geometry, ed. by F. Dillen, L.Verstraelen, Handbook of Differential Geometry, Vol. I, Elsevier Science, Amsterdam, (2000), 905–961.
  • [26] Vos, P. W., Fundamental equations for statistical submanifolds with applications to the Bartlett correction. Ann. Inst. Stat. Math. 41(3) (1989), 429–450.
  • [27] Wintgen, P., Sur l’inégalité de Chen-Willmore. C. R. Acad. Sci., Paris Sér. A-B 288 (1979), A993-A995.
  • [28] Yano, K. and Kon, M., Structures on manifolds, Series in Pure Mathematics, 3. World Scientific Publishing Co., Singapore, 1984.
  • [29] Todjihounde, L., Dualistic structures on warped product manifolds. Diff.Geom.-Dyn. Syst., 8 (2006), 278-284.
  • [30] Yazla, A., Küpeli Erken, ˙I. and Murathan, C., Almost cosymplectic statistical manifolds. Quaestiones Mathematicae in press, https://doi.org/10.2989/16073606.2019.1576069.

A Generalized Wintgen Inequality for Legendrian Submanifolds in Almost Kenmotsu Statistical Manifolds

Year 2019, Volume: 12 Issue: 1, 43 - 56, 27.03.2019

Ruken Görünüş İrem Küpeli Erken , Aziz Yazla Cengizhan Murathan

Abstract

Main interest of the present paper is to obtain the generalized Wintgen inequality for Legendrian submanifolds in almost Kenmotsu statistical manifolds.

Keywords

Statistical Manifolds Cosymplectic Manifolds Kenmotsu Manifolds Wintgen inequality Legendrian submanifold

References

  • [1] Amari, S., Differential-Geometrical Methods in Statistics. Lecture Notes in Statistics. 28 Springer, Berlin 1985.
  • [2] Aydın, M. E., Mihai, A. and Mihai, I., Some inequalities on submanifolds in statistical manifolds of constant curvature. Filomat 29(3) (2015), 465-477.
  • [3] Aydın, M.E., Mihai, A. and Mihai I., Generalized Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature. Bull. Math. Sci. 7(1) (2017), 155-166.
  • [4] Aydın, M.E. and Mihai I., Wintgen inequality for statistical surfaces. Math. Inequal. Appl. 22(1)(2019), 123–132.
  • [5] Aytimur, H. and Özgür, C., Inequalities for submanifolds in statistical manifolds of quasi-constant curvature. Ann. Polon. Math. 121 (2018), no. 3, 197–215.
  • [6] Blair D. E., Riemannian Geometry of Contact and Symplectic Manifolds. Boston. Birkhâuser 2002.
  • [7] Boyom, M. N., Aquib, M., Shahid M.H. and Jamali,M., Generalized Wintgen Type Inequality for Lagrangian Submanifolds in Holomorphic Statistical Space Forms. Frank Nielsen  Frédéric Barbaresco (Eds.) Geometric Science of Information Third International Conference, GSI 2017 Paris, France, November 7–9, 2017.
  • [8] Carriazo, A. and Perez-Garcia, M.J., Slant submanifolds in neutral almost contact pseudo-metric manifolds. Differ. Geom. Appl. 54 (2017), 71–80.
  • [9] Chen, B. Y., Mean curvature and shape operator of isometric immersions in real-space forms. Glasgow Math. J. 38 (1996), 87-97.
  • [10] Chen, Q. and Cui, Q., Normal scalar curvature and a pinching theorem in Sm  R and Hm  R. Science China Math. 54(9) (2011), 1977- 1984.
  • [11] De Smet, P. J., Dillen, F., Verstraelen, L. and Vrancken, L., A pointwise inequality in submanifold theory. Arch. Math. (Brno) 35 (1999), 115-128.
  • [12] Dillen, F., Fastenakels, J. and Van der Veken, J., Remarks on an inequality involving the normal scalar curvature. Pure and Applied Differential Geometry-PADGE 2007, 83-92, Ber. Math., Shaker Verlag, Aachen, 2007.
  • [13] Furuhata, H., Hypersurfaces in statistical manifolds. Diff. Geom. Appl. 27 (2009), 420-429.
  • [14] Furuhata, H., Hasegawa, I., Okuyama, Y. and Sato, K., Kenmotsu statistical manifolds and warped product. J. Geom. 108 (2017), 1175–1191.
  • [15] Ge, J. and Tang, Z., A proof of the DDVV conjecture and its equality case. Pacific J. Math. 237 (2008), 87-95.
  • [16] Kenmotsu, K., A class of contact Riemannian manifold. Tohoku Math. Journal 24 (1972), 93-103.
  • [17] Lauritzen, S., Statistical manifolds. In: Amari, S., Barndorff-Nielsen, O., Kass, R., Lauritzen, S., Rao, C.R. (eds.) Differential Geometry in Statistical Inference, 10, 163–216. IMS Lecture NotesInstitute of Mathematical Statistics, Hayward 1987.
  • [18] Lawn, M. and Ortega, M., A fundamental theorem for hypersurfaces in semi-Riemannian warped products. J. Geom. Phys. 90 (2015), 55-70. [19] Lu, Z., Normal scalar curvature conjecture and its applications. J. Funct. Analysis 261 (2011), 1284-1308.
  • [20] Mihai, I., On the generalizedWintgen inequality for Lagrangian submanifolds in complex space forms. Nonlinear Anal. 95 (2014), 714-720.
  • [21] Mihai, I., On the generalized Wintgen inequality for Legendrian submanifolds in Sasakian space forms. Tohoku Math. J. 69 (2017), 43-53.
  • [22] Murathan, C. and ¸Sahin, B., A study of Wintgen like inequality for submanifolds in statistical warped product manifolds. J. Geom. 109 (2018), 18 pp.
  • [23] Opozda, B., A sectional curvature for statistical structures. Linear Algebra Appl. 497 (2016), 134–161.
  • [24] Roth, J., A DDVV Inequality for submanifolds of warped products. Bull. Aust. Math. Soc. 95 (2017), 495–499.
  • [25] Simon, U., Affine Differential Geometry, ed. by F. Dillen, L.Verstraelen, Handbook of Differential Geometry, Vol. I, Elsevier Science, Amsterdam, (2000), 905–961.
  • [26] Vos, P. W., Fundamental equations for statistical submanifolds with applications to the Bartlett correction. Ann. Inst. Stat. Math. 41(3) (1989), 429–450.
  • [27] Wintgen, P., Sur l’inégalité de Chen-Willmore. C. R. Acad. Sci., Paris Sér. A-B 288 (1979), A993-A995.
  • [28] Yano, K. and Kon, M., Structures on manifolds, Series in Pure Mathematics, 3. World Scientific Publishing Co., Singapore, 1984.
  • [29] Todjihounde, L., Dualistic structures on warped product manifolds. Diff.Geom.-Dyn. Syst., 8 (2006), 278-284.
  • [30] Yazla, A., Küpeli Erken, ˙I. and Murathan, C., Almost cosymplectic statistical manifolds. Quaestiones Mathematicae in press, https://doi.org/10.2989/16073606.2019.1576069.
There are 29 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

Ruken Görünüş This is me

İrem Küpeli Erken

Aziz Yazla This is me

Cengizhan Murathan

Publication Date March 27, 2019
Published in Issue Year 2019 Volume: 12 Issue: 1

Cite

APA Görünüş, R., Erken, İ. K., Yazla, A., Murathan, C. (2019). A Generalized Wintgen Inequality for Legendrian Submanifolds in Almost Kenmotsu Statistical Manifolds. International Electronic Journal of Geometry, 12(1), 43-56.
AMA Görünüş R, Erken İK, Yazla A, Murathan C. A Generalized Wintgen Inequality for Legendrian Submanifolds in Almost Kenmotsu Statistical Manifolds. Int. Electron. J. Geom. March 2019;12(1):43-56.
Chicago Görünüş, Ruken, İrem Küpeli Erken, Aziz Yazla, and Cengizhan Murathan. “A Generalized Wintgen Inequality for Legendrian Submanifolds in Almost Kenmotsu Statistical Manifolds”. International Electronic Journal of Geometry 12, no. 1 (March 2019): 43-56.
EndNote Görünüş R, Erken İK, Yazla A, Murathan C (March 1, 2019) A Generalized Wintgen Inequality for Legendrian Submanifolds in Almost Kenmotsu Statistical Manifolds. International Electronic Journal of Geometry 12 1 43–56.
IEEE R. Görünüş, İ. K. Erken, A. Yazla, and C. Murathan, “A Generalized Wintgen Inequality for Legendrian Submanifolds in Almost Kenmotsu Statistical Manifolds”, Int. Electron. J. Geom., vol. 12, no. 1, pp. 43–56, 2019.
ISNAD Görünüş, Ruken et al. “A Generalized Wintgen Inequality for Legendrian Submanifolds in Almost Kenmotsu Statistical Manifolds”. International Electronic Journal of Geometry 12/1 (March2019), 43-56.
JAMA Görünüş R, Erken İK, Yazla A, Murathan C. A Generalized Wintgen Inequality for Legendrian Submanifolds in Almost Kenmotsu Statistical Manifolds. Int. Electron. J. Geom. 2019;12:43–56.
MLA Görünüş, Ruken et al. “A Generalized Wintgen Inequality for Legendrian Submanifolds in Almost Kenmotsu Statistical Manifolds”. International Electronic Journal of Geometry, vol. 12, no. 1, 2019, pp. 43-56.
Vancouver Görünüş R, Erken İK, Yazla A, Murathan C. A Generalized Wintgen Inequality for Legendrian Submanifolds in Almost Kenmotsu Statistical Manifolds. Int. Electron. J. Geom. 2019;12(1):43-56.