Inequalities involving k-Chen invariants for submanifolds of Riemannian product manifolds

Mehmet Gülbahar; Mukut Mani Trıpathı; Erol Kılıç

doi:10.31801/cfsuasmas.430856

Research Article

Inequalities involving k-Chen invariants for submanifolds of Riemannian product manifolds

Year 2019, Volume: 68 Issue: 1, 466 - 483, 01.02.2019

Mehmet Gülbahar Mukut Mani Trıpathı Erol Kılıç

https://doi.org/10.31801/cfsuasmas.430856

Abstract

An optimal inequality involving the scalar curvatures, the mean curvature and the k-Chen invariant is established for Riemannian submanifolds. Particular cases of this inequality is reported. Furthermore, this inequality is investigated on submanifolds, namely slant, F-invariant and F-anti invariant submanifolds of an almost constant curvature manifold.

Keywords

Curvature, Riemannian submanifolds, almost constant curvature manifold

References

Adati, T., Submanifolds of an almost product manifold, Kodai Math. J. (1981), 4, 327-343.
Atçeken, M., Slant submanifolds of a Riemannian product manifold, Acta Math. Sci. Ser. B. Engl. Ed. (2010), 30(1), 215-224.
Bejancu, A., CR submanifolds of a Kaehler manifold I., Proc. Amer. Math. Soc. (1978), 69(1), 135-142.
Chen, B.-Y., A Riemannian invariant and its applications to submanifold theory, Festschrift dedicated to Katsumi Nomizu on his 70th birthday (Leuven, 1994; Brussels, 1994), Results Math. (1995), 27(1-2), 17-26.
Chen, B.-Y., Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions, Glasgow Math. J. (1999), 41, 33-41.
Chen, B.-Y., Riemannian DNA, inequalities and their applications, Tamkang J. Sci. and Eng. (2000), 3, 123-130.
Chen, B.-Y., Riemannian submanifolds, in Handbook of Differential Geometry, Vol. I, eds. F. Dillen and L. Verstraelen, North Holland, Amsterdam, 2000, 187-418.
Chen, B.-Y., On Ricci curvature of isotropic and Langrangian submanifolds in complex space forms, Arch. Math. (Basel) (2000), 74, 154-160.
Chen, B.-Y., Some new obstructions to minimal and Lagrangian isometric immersions, Japan. J. Math. (N.S.) (2000), 26(1), 105-127.
Chen, B.-Y., Pseudo-Riemannian Geometry, δ-invariants and applications, World Scientific, 2011.
Gülbahar, M., Kılıc, E. and Saraçoglu Çelik S., Special proper pointwise slant surfaces of a locally product Riemannian manifold, Turk. J. Math. (2015), 39, 884-899.
Kılıç, E., Tripathi, M. M. and Gülbahar, M., Chen-Ricci inequalities for submanifolds of Riemannian and Kaehlerian product manifolds, Ann. Polon. Math. (2016), 116(1), 37-56.
Liu, X. and Shao, F.-M., Skew semi-invariant submanifolds of a locally product manifold, Portugal. Math. (1999), 56(3), 319-327.
Sahin, B., Slant submanfiolds of an almost product Riemannian manifold, J. Korean Math. Soc. (2006), 43(4), 717-732.
Singer, I. M. and Thorpe, J. A., The curvature of 4-dimensional Einstein spaces, Global Analysis, Princeton University Press, 355-365, 1969.
Tachibana, S., Some theorems on locally product Riemannian spaces, Tohoku Math. J. (1960), 12, 281-293.
Tripathi, M. M., Almost semi-invariant submanifolds of a parallel ε-framed metric manifold, Ganita (1997), 48(1) 57-70. Yano, K., Differential geometry on complex and almost complex spaces, International Series of Monographs in Pure and Applied Mathematics 49. A Pergamon Press Book. The Macmillan Co., New York, 1965.
Yano, K., and Kon, M., Submanifolds of Kaehlerian product manifolds, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia. (1979), 15(5), 267-292.
Yano, K. and Kon, M., Structures on manifolds, Series in Pure Mathematics, World Scientific Publishing Co., Singapore, 1984.

Year 2019, Volume: 68 Issue: 1, 466 - 483, 01.02.2019

Mehmet Gülbahar Mukut Mani Trıpathı Erol Kılıç

https://doi.org/10.31801/cfsuasmas.430856

Abstract

References

Adati, T., Submanifolds of an almost product manifold, Kodai Math. J. (1981), 4, 327-343.
Atçeken, M., Slant submanifolds of a Riemannian product manifold, Acta Math. Sci. Ser. B. Engl. Ed. (2010), 30(1), 215-224.
Bejancu, A., CR submanifolds of a Kaehler manifold I., Proc. Amer. Math. Soc. (1978), 69(1), 135-142.
Chen, B.-Y., A Riemannian invariant and its applications to submanifold theory, Festschrift dedicated to Katsumi Nomizu on his 70th birthday (Leuven, 1994; Brussels, 1994), Results Math. (1995), 27(1-2), 17-26.
Chen, B.-Y., Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions, Glasgow Math. J. (1999), 41, 33-41.
Chen, B.-Y., Riemannian DNA, inequalities and their applications, Tamkang J. Sci. and Eng. (2000), 3, 123-130.
Chen, B.-Y., Riemannian submanifolds, in Handbook of Differential Geometry, Vol. I, eds. F. Dillen and L. Verstraelen, North Holland, Amsterdam, 2000, 187-418.
Chen, B.-Y., On Ricci curvature of isotropic and Langrangian submanifolds in complex space forms, Arch. Math. (Basel) (2000), 74, 154-160.
Chen, B.-Y., Some new obstructions to minimal and Lagrangian isometric immersions, Japan. J. Math. (N.S.) (2000), 26(1), 105-127.
Chen, B.-Y., Pseudo-Riemannian Geometry, δ-invariants and applications, World Scientific, 2011.
Gülbahar, M., Kılıc, E. and Saraçoglu Çelik S., Special proper pointwise slant surfaces of a locally product Riemannian manifold, Turk. J. Math. (2015), 39, 884-899.
Kılıç, E., Tripathi, M. M. and Gülbahar, M., Chen-Ricci inequalities for submanifolds of Riemannian and Kaehlerian product manifolds, Ann. Polon. Math. (2016), 116(1), 37-56.
Liu, X. and Shao, F.-M., Skew semi-invariant submanifolds of a locally product manifold, Portugal. Math. (1999), 56(3), 319-327.
Sahin, B., Slant submanfiolds of an almost product Riemannian manifold, J. Korean Math. Soc. (2006), 43(4), 717-732.
Singer, I. M. and Thorpe, J. A., The curvature of 4-dimensional Einstein spaces, Global Analysis, Princeton University Press, 355-365, 1969.
Tachibana, S., Some theorems on locally product Riemannian spaces, Tohoku Math. J. (1960), 12, 281-293.
Tripathi, M. M., Almost semi-invariant submanifolds of a parallel ε-framed metric manifold, Ganita (1997), 48(1) 57-70. Yano, K., Differential geometry on complex and almost complex spaces, International Series of Monographs in Pure and Applied Mathematics 49. A Pergamon Press Book. The Macmillan Co., New York, 1965.
Yano, K., and Kon, M., Submanifolds of Kaehlerian product manifolds, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia. (1979), 15(5), 267-292.
Yano, K. and Kon, M., Structures on manifolds, Series in Pure Mathematics, World Scientific Publishing Co., Singapore, 1984.

There are 19 citations in total.

Details

Primary Language	English
Journal Section	Review Articles
Authors	Mehmet Gülbahar 0000-0001-6950-7633 Mukut Mani Trıpathı 0000-0002-6113-039X Erol Kılıç 0000-0001-7536-0404
Publication Date	February 1, 2019
Submission Date	October 10, 2017
Acceptance Date	February 20, 2018
Published in Issue	Year 2019 Volume: 68 Issue: 1

Cite

APA	Gülbahar, M., Trıpathı, M. M., & Kılıç, E. (2019). Inequalities involving k-Chen invariants for submanifolds of Riemannian product manifolds. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(1), 466-483. https://doi.org/10.31801/cfsuasmas.430856
AMA	Gülbahar M, Trıpathı MM, Kılıç E. Inequalities involving k-Chen invariants for submanifolds of Riemannian product manifolds. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. February 2019;68(1):466-483. doi:10.31801/cfsuasmas.430856
Chicago	Gülbahar, Mehmet, Mukut Mani Trıpathı, and Erol Kılıç. “Inequalities Involving K-Chen Invariants for Submanifolds of Riemannian Product Manifolds”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68, no. 1 (February 2019): 466-83. https://doi.org/10.31801/cfsuasmas.430856.
EndNote	Gülbahar M, Trıpathı MM, Kılıç E (February 1, 2019) Inequalities involving k-Chen invariants for submanifolds of Riemannian product manifolds. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 1 466–483.
IEEE	M. Gülbahar, M. M. Trıpathı, and E. Kılıç, “Inequalities involving k-Chen invariants for submanifolds of Riemannian product manifolds”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 68, no. 1, pp. 466–483, 2019, doi: 10.31801/cfsuasmas.430856.
ISNAD	Gülbahar, Mehmet et al. “Inequalities Involving K-Chen Invariants for Submanifolds of Riemannian Product Manifolds”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68/1 (February 2019), 466-483. https://doi.org/10.31801/cfsuasmas.430856.
JAMA	Gülbahar M, Trıpathı MM, Kılıç E. Inequalities involving k-Chen invariants for submanifolds of Riemannian product manifolds. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68:466–483.
MLA	Gülbahar, Mehmet et al. “Inequalities Involving K-Chen Invariants for Submanifolds of Riemannian Product Manifolds”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 68, no. 1, 2019, pp. 466-83, doi:10.31801/cfsuasmas.430856.
Vancouver	Gülbahar M, Trıpathı MM, Kılıç E. Inequalities involving k-Chen invariants for submanifolds of Riemannian product manifolds. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68(1):466-83.