Estimates of B.-Y. Chen’s δˆ-Invariant in Terms of Casorati Curvature and Mean Curvature for Strictly Convex Euclidean Hypersurfaces

Abstract

B.-Y. Chen’s δˆ-invariants can be estimated in function of other curvature terms through an algebraic

process using the AM-GM and AM-QM inequalities. This procedure works on strictly convex

smooth hypersurfaces lying in an Euclidean ambient space, and the estimates relate some  δˆ-

invariants to Germain’s mean curvature and Casorati curvature. As a consequence, we obtain a

new string of inequalities in the geometry of strictly convex smooth hypersurfaces.

Keywords

• principal curvatures,Casorati curvature,mean curvature,smooth hypersurfaces,AM QM inequality

References

1. [1] Brubaker, N.D.; Suceava, B.D., A Geometric Interpretation of Cauchy-Schwarz Inequality in Terms of Casorati Curvature. Int. Electron. J. Geom., 11(2018), 48-51.
2. [2] Brzycki, B.; Giesler, M.D.; Gomez, K.; Odom L.H.; and Suceava, B.D., A ladder of curvatures for hypersurfaces in the Euclidean ambient space. Houston Journal of Mathematics, 40(2014). pp. 1347-1356.
3. [3] Casorati, Felice, Mesure de la courbure des surfaces suivant l’idée commune. Ses rapports avec les mesures de courbure gaussienne et moyenne, Acta Math., 14(1)(1890), 95–110.
4. [4] Chen, Bang-Yen, Geometry of submanifolds, M. Dekker, New York, 1973.
5. [5] Chen, Bang-Yen, Geometry of submanifolds and its applications, Science University of Tokyo, 1981.
6. [6] Chen, Bang-Yen, Some pinching and classification theorems for minimal submanifolds. Arch. Math., 60(1993), 568-578.
7. [7] Chen, Bang-Yen, A Riemannian invariant and its applications to submanifold theory. Results Math., 27(1995), 17-26.
8. [8] Chen, Bang-Yen, Some new obstructions to minimal and Lagrangian isometric immersions. Japanese J. Math., 26(2000), 105-127.

9. [9] Chen, Bang-Yen, A series of Kählerian invariants and their applications to Kählerian geometry. Beiträge Algebra Geom. ,42(2001), 165-178. [10] Chen, Bang-Yen, Pseudo-Riemannian submanifolds, -invariants and Applications, World Scientific, 2011.
10. [11] Conley, C. T. R.; Etnyre, R.; Gardener, B.; Odom L.H.; and Suceava, B.D., New Curvature Inequalities for Hypersurfaces in the Euclidean Ambient Space. Taiwanese Journal of Mathematics, 17(2013), 885-895.
11. [12] De Smet, P. J.; Dillen, F.; Verstraelen, L.; Vrancken, L., A pointwise inequality in submanifold theory. Arch. Math. (Brno), 35(1999), no. 2, 115–128.
12. [13] Decu, S.; Haesen, S.; and Verstraelen, L., Optimal inequalities involving Casorati curvatures. Bull. Transylv. Univ. Bra¸sov Ser. B , 14(2007), 85–93.
13. [14] Decu, S.; Haesen, S.; Verstraelen, L.; Vîlcu, G.-E., Curvature Invariants of Statistical Submanifolds in Kenmotsu Statistical Manifolds of Constant -Sectional Curvature. Entropy, 20(2018) 20, 529.
14. [15] Dillen, F.; Fastenakels, J.; Van der Veken, J., A pinching theorem for the normal scalar curvature of invariant submanifolds. J. Geom. Phys., 57(2007), no. 3, 833–840.
15. [16] doCarmo, Manfredo P., Riemannian Geometry, Birkhäuser, 1992.
16. [17] Gauss, C.F. , Disquisitiones circa superficies curvas, Typis Dieterichianis, Goettingen, 1828.
17. [18] Germain, S., Mémoire sur la courbure des surfaces. Journal für die reine und andewandte Mathematik, Herausgegeben von A. L. Crelle, Siebenter Band, pp. 1–29, Berlin, 1831.
18. [19] Haesen, S; Kowalczyk, D.; Verstraelen, L., On the extrinsic principal directions of Riemannian submanifolds. Note Mat., 29(2009), no. 2, 41–53.
19. [20] Koenderink, J.; van Doorn, A.; Pont, S., Shading, a View from the Inside, Seeing and Perceiving 25(2012), 303–338.
20. [21] Lee, C. W.; Lee, J. W.; Vîlcu, G.-E., Optimal inequalities for the normalized δ-Casorati curvatures of submanifolds in Kenmotsu space forms. Adv. Geom., 17(2017), no. 3, 355–362.
21. [22] Lu, Z., Normal scalar curvature conjecture and its applications. J. Funct. Anal., 261(2011), 1284–1308.
22. [23] Mihai, I., On the generalized Wintgen inequality for Legendrian submanifolds in Sasakian space forms. Tohoku Math. J., 69(2017), no. 1, 43–53.
23. [24] Suceava, B. D., The spread of the shape operator as conformal invariant. J. Inequal. Pure Appl. Math, 4(2003), issue 4, article 74.
24. [25] Suceava, B. D., Fundamental inequalities and strongly minimal submanifolds. In Duggal, Krishan L. (ed.) et al., Recent advances in Riemannian and Lorentzian geometries. Proceedings of the special session of the annual meeting of the American Mathematical Society, Baltimore, MD, USA, January 15-18, 2003, American Mathematical Society (AMS) Contemporary Mathematics 337, 155–170.
25. [26] Suceava, B. D., The amalgamatic curvature and the orthocurvatures of three dimensional hypersurfaces in E^4. Publicationes Mathematicae, 87(2015), no. 1-2, 35–46.
26. [27] Suceava, B.D., A Geometric Interpretation of Curvature Inequalities on Hypersurfaces via Ravi Substitutions in the Euclidean Plane. Math Intelligencer 40(2018), 50–54.
27. [28] Suceava, B.D.; Vajiac, M. B. Remarks on Chen’s fundamental inequality with classical curvature invariants in Riemannian spaces. An. Ştiin¸t. Univ. Al. I. Cuza Iaşi. Mat. (N.S.), 54(2008), no. 1, 27–37.
28. [29] Verstraelen, L., Geometry of submanifolds I. The first Casorati curvature indicatrices. Kragujevac J. Math., 37(2013), 5–23.
29. [30] Verstraelen, L., A Concise Mini History of Geometry. Kragujevac J. Math., 38(2013), 5–21.
30. [31] Vîlcu, G.-E., On Chen invariants and inequalities in quaternionic geometry. J. Inequal. Appl., 2013: 66, 14 pp.