Hineva and Chen-Ricci Inequalities for Ricci Curvature of Submanifolds of Product Generalized Sasakian Space Forms

Abstract

The sectional curvature, Ricci curvature, and scalar curvature for a product generalized Sasakian space form are obtained. Furthermore, the Chen-Ricci inequality and the Hineva inequality are established for submanifolds of a product generalized Sasakian space form, including product Sasakian, product cosymplectic, and product Kenmotsu space forms. The equality cases are also discussed.

Keywords

• Kulkarni-Nomizu tensor field
• algebraic Gauss equation
• Chen-Ricci inequality
• Hineva inequality
• quasi-umbilical submanifold
• product generalized Sasakian space form
• product Sasakian space form
• product cosymplectic space form
• product Kenmotsu space form

References

1. Adati T.: Submanifolds of an almost product Riemannian manifold, Kodai Math. J. 4 (1981), no. 2, 327–343.
2. Alegre P., Blair D. E., Carriazo A.: Generalized Sasakian space forms, Israel J. Math. 141 , 157–183 (2004).
3. Bejancu A.: CR submanifolds of a Kaehler manifold I, Proc. Amer. Math. Soc. 69 no. 1, 135–142 (1978).
4. Beem J. K., Ehrlich P. E., Easley K. L.: Global Lorentzian geometry, Second edition, Monographs and Textbooks in Pure and Applied Mathematics, 202. Marcel Dekker, Inc., New York, 1996, xiv+635
5. Besse A. L.: Einstein Manifolds, Reprint of the 1987 edition. Classics in Mathematics, Springer-Verlag, Berlin, 2008.
6. Bolton J., Dillen F., Fastenakels J., Vrancken L.: A best possible inequality for curvature-like tensor fields, Math. Inequal. Appl. 12 no. 3, 663–681 (2009).
7. Blair D. E.: Riemannian geometry of contact and symplectic manifolds, Second edition, Progress in Mathematics, 203, Birkhäuser Boston, Ltd., Boston, MA, 2010.
8. Capursi M.: Some remarks on the product of two almost contact manifolds, An. Stiint. Univ. “AI. I. Cuza" Iasi, Sect. Ia Mat. 30 no. 1, 75–79 (1984).

9. Capursi M., Ianus S.: Complex hypersurfaces of the product of two cosymplectic manifolds I, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 28 (76) no. 4, 299–310(1984).
10. Capursi M., Ianus S.: Complex hypersurfaces of the product of two cosymplectic manifolds II, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 29 (77) no. 3, 215–224 (1985).
11. Capursi M., Ianus S.: On the scalar curvature of a complex hypersurface of the product of two cosymplectic space forms, Tensor (N.S.) 42 no. 2, 131–136 (1985).
12. Chen B. Y.: Geometry of submanifolds, Pure and Applied Mathematics, No. 22. Marcel Dekker, Inc., New York, 1973.
13. Chen B. Y.: Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions, Glasgow Math. J. 41 , 33–41 (1999).
14. Chen B. Y.: Riemannian submanifolds, Handbook of differential geometry, Vol. I, 187–418, North-Holland, Amsterdam, 2000.
15. [15] Chen B. Y., Dillen F., Verstraelen L.: δ-invariants and their applications to centroaffine geometry, Differential Geom. Appl. 22 no. 3, 341–354 (2005).
16. Chen B. Y., Blaga A. M.: Recent developments on Chen-Ricci inequalities in differential geometry, Geometry of submanifolds and applications, 1–61, Infosys Sci. Found. Ser., Infosys Sci. Found. Ser. Math. Sci., Springer, Singapore, 2024. MR4759264.
17. Choudhary M. A.: Totally real submanifolds of Kaehler product manifolds, J. Egyptian Math. Soc. 24 no. 2, 258–262 (2016).
18. Cvetic M., Youm D.: Rotating intersectingM-branes, Nuclear Phys. B 499 no. 1–2, 253–282 (1997).
19. Deng S.: An improved Chen-Ricci inequality, Int. Electron. J. Geom. 2 (2), 39–45 (2009).
20. Gallot S., Hulin D., Lafontaine J.: Riemannian geometry, Universitext, Springer-Verlag, Berlin, 1987.
21. Hineva S. T.: Submanifolds and the second fundamental tensor, Global differential geometry and global analysis 1984 (Berlin, 1984), 194–203, Lecture Notes in Math., 1156, Springer, Berlin, 1985.
22. Hineva S. T., Beltchev E.: A note on the submanifolds of a δ-pinched manifold, Annuaire Univ. Sofia Fac. Math. Inform. 81 (1987), no. 1, 59–65 (1994).
23. Hineva S. T.: Inequalities for sectional curvature and the Ricci curvature of submanifolds, Mathematics and mathematical education (Slanchev Bryag, 1990), 7–8, Publ. House Bulgar. Acad. Sci., Sofia, 1990.
24. Hineva S. T.: Submanifolds for which a lower bound of the Ricci curvature is achieved, J. Geom. 88 no. 1–2, 53–69 (2008).
25. Hong S., Tripathi M. M.: On Ricci curvature of submanifolds, Int. J. Pure Appl. Math. Sci. 2 no. 2, 227–245 (2005).
26. Hong S., Tripathi M. M.: On Ricci curvature of submanifolds of generalized Sasakian space forms, Int. J. Pure Appl. Math. Sci. 2 no. 2, 173– 201(2005).
27. Holm M.: New insights in brane and Kaluza-Klein theory through almost product structures, arXiv:hep-th/9812168 (1998).
28. Holm M., Sandström N.: Curvature relations in almost product manifolds, arXiv:hep-th/9904099 (1999).
29. Kenmotsu K.: A class of almost contact Riemannian manifolds, Tohoku Math. J. 24, 93–103 (1972).
30. Kılıç E., Tripathi M. M., Gülbahar M.: Chen-Ricci inequalities for submanifolds of Riemannian and Kaehlerian product manifolds, Ann. Polon. Math. 116 no. 1, 37–56 (2016).
31. Kulkarni R. S.: Curvature and metric, Ph.D. Thesis, Harvard University, Cambridge, Mass., 1967.
32. Kulkarni R. S.: Curvature structures and conformal transformations, Bull. Amer. Math. Soc. 75, 91–94(1969).
33. Liu X., Shao F. M.: Skew semi-invariant submanifolds of a locally product manifold, Portugal. Math. 56 no. 3, 319–327 (1999).
34. Liu X., Xiao B.: Minimal submanifolds in certain types of Kaehler product manifold, Southeast Asian Bull. Math. 43 ( no. 1, 79–100 2019).
35. Matsuyama Y.: Complex hypersurfaces of the product of two complex space forms, Kodai Math. Sem. Rep. 28 no. 2–3, 144–149 (1976).
36. Matsuyama Y.: Complex hypersurfaces of Pn(C) × Pn(C), Tohoku Math. J. (2) 28 no. 4, 541–552 (1976).
37. Morimoto A.: On normal almost contact structures, J. Math. Soc. Japan 15, 420–436 (1963).
38. Mihai I., R˘adulescu I. N.: An improved Chen-Ricci inequality for Kaehlerian slant submanifolds in complex space forms, Taiwanese J. Math. 16, 761–770 (2012).
39. Mihai A., R˘adulescu I. N.: Scalar and Ricci curvatures of special contact slant submanifolds in Sasakian space forms, Adv. Geom. 14, 147–159 (2014).
40. Mihai A.: Inequalities on the Ricci curvature, J. Math. Inequal. 9, 811–822(2015).
41. Mihai I., Radulescu I. N.: An improved Chen-Ricci inequality for Legendrian submanifolds in Sasakian space forms, J. Adv. Math. Stud. 4 , 51–58 (2011).
42. Nash J. F. Jr.: The imbedding problem for Riemannian manifolds, Ann. of Math. 63, 20–63 (1956).
43. Oprea T. V.: On a geometric inequality, arXiv:math.DG/0511088v1, 3 November 2005.
44. Oprea T. V.: Ricci curvature of Lagrangian submanifolds in complex space forms, Math. Inequal. Appl. 13 no. 4, 851–858(2010).
45. Sahin B., Keles S.: Slant submanifolds of Kaehler product manifolds, Turkish J. Math. 31 no. 1, 65–77 (2007).
46. Shahid M. H.: CR-submanifolds of Kaehlerian product manifold, Indian J. Pure Appl. Math. 23 no. 12, 873–879 (1992).
47. Singh K. D., Tripathi M. M.: On normal (e1, e2, r)ac structure, Ganita 40 no. 1–2, 101–112 (1990).
48. Tachibana S.: Some theorems on locally Riemannian spaces, Tohoku Math. J. 12, 281–292 (1960).
49. Tripathi M. M., Singh K. D.: Almost semi-invariant submanifolds of an ϵ-framed metric manifold, Demonstratio Math. 29 no. 2, 413–426 (1996).
50. Tripathi M. M.: Almost semi-invariant submanifolds of a parallel ϵ-framed metric manifold, Ganita 48 no. 1, 57–70 (1997).
51. Tripathi M. M.: Chen-Ricci inequality for submanifolds of contact metric manifolds, J. Adv. Math. Stud. 1 no. 1–2, 111–134 (2008).
52. Tripathi M. M.: Improved Chen-Ricci inequality for curvature-like tensors and its applications, Differential Geom. Appl. 29 no. 5, 685–698 (2011).
53. Tripathi M. M.: Kulkarni Nomizu tensor fields, Geometry, groups and mathematical philosophy, Contemp. Math. Amer. Math. Soc. 811, 259–278 (2025).
54. Tripathi M. M.: Handbook of the differential geometry of space forms, Series in Algebraic and Differential Geometry 4, World Scientific, London, United Kingdom, (in press)2026.
55. Verma K. K., Mihai A., Mihai I., Tripathi M. M.: Hineva inequality for Ricci curvature of Kulkarni-Nomizu tensor fields and its applications, (to appear) (2026).
56. Yano K.: On Walker differentiation in almost product or almost complex spaces, Nederl. Akad. Wetensch. Proc. Ser. A 61 / Indag. Math. 20, 573–580 (1958).
57. 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.
58. Yano K., Kon M.: Submanifolds of Kaehlerian product manifolds, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia 15 (8) , no. 5, 267–292 (1979).
59. Yano K., Kon M.: Structures on manifolds, Series in Pure Mathematics, 3. World Scientific Publishing Co., Singapore, 1984.