A Characterization of Cylinders and an Estimate for Mean Curvature of Convex Euclidean Hypersurfaces Satisfying a Cylindrical Condition

Yıl 2019, Cilt: 12 Sayı: 1, 57 - 60, 27.03.2019

Leonard M. Giugiuc Bogdan D. Suceava

https://doi.org/10.36890/iejg.545755

Öz

We show that a curvature condition on the Gauss-Kronecker curvature and scalar curvature of a

convex smooth hypersurface lying in the four dimensional Euclidean space yields a lower bound

for the mean curvature. The curvature condition we investigate is suggested by the local geometry

of cylinders in the four dimensional Euclidean space.

Anahtar Kelimeler

principal curvatures, mean curvature, scalar curvature, Gauss-Kronecker curvature, smooth hypersurfaces

Kaynakça

• [1] Bonnet, O., Sur quelque propriétés des lignes géodésiques. Comptes rendus de l’Academie des Sciences 11 (1855), 1311–1313.
• [2] Brubaker, N.D. and Suceava, B.D., A Geometric Interpretation of Cauchy-Schwarz Inequality in Terms of Casorati Curvature. Intern. Elec. J. Geom. 11 (2018), 48–51.
• [3] Brubaker, N.D., Camero, J., Rocha Rocha O., Soto, R. and Suceav˘a, B.D., A Curvature Invariant Inspired by Leonhard Euler’s Inequality R ≥ 2r. Forum Geometricorum 18 (2018) 119–127.
• [4] Brzycki, B., Giesler, M.D., Gomez, K., Odom L.H. and Suceav˘a, B.D., A ladder of curvatures for hypersurfaces in the Euclidean ambient space. Houston J. Math. 40 (2014) 1347–1356.
• [5] Calabi, E., On Ricci curvatures and geodesics. Duke Math. J. 34 (1967), 667–676.
• [6] Chen, B.-Y., Some pinching and classification theorems for minimal submanifolds. Arch. Math. 60 (1993), 568-578.
• [7] Chen, B.-Y., A Riemannian invariant and its applications to submanifold theory. Results Math. 27 (1995), 17-26.
• [8] Chen, B.-Y., Some new obstructions to minimal and Lagrangian isometric immersions. Japanese J. Math. 26 (2000), 105-127.
• [9] Chen, B.-Y., Pseudo-Riemannian submanifolds, δ-invariants and Applications. World Scientific, 2011.
• [10] 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 J. Math. 17 (2013), 885–895.
• [11] Dillen, F., Fastenakels, J. and Van der Veken, J., A pinching theorem for the normal scalar curvature of invariant submanifolds. J. Geom. Phys. 57 (2007), no. 3, 833–840.
• [12] doCarmo, Manfredo P., Riemannian Geometry. Birkhäuser, 1992.
• [13] Galloway, G.J., A Generalization of Myers Theorem and an application to relativistic cosmology. J. Diff. Geom. 14 (1979), 105–116.
• [14] Gauss, C.F. , Disquisitiones circa superficies curvas. Typis Dieterichianis, Goettingen, 1828.
• [15] Germain, S., Mémoire sur la courbure des surfaces. J. Reine Angew. Math. 8 (1832), 280–297.
• [16] Giugiuc, L.M., Problem 11911. American Mathematical Monthly 123 (2016), p. 504.
• [17] Myers, S.B., Riemmannian manifolds with positive curvature. Duke Math. J. 8 (1941), 401–404.
• [18] Suceav˘a, B. D., The spread of the shape operator as conformal invariant. J. Inequal. Pure Appl. Math. 4 (2003), article 74.
• [19] Suceav˘a, B. D., The amalgamatic curvature and the orthocurvatures of three dimensional hypersurfaces in E4. Publicationes Mathematicae 87 (2015), no. 1-2, 35–46.
• [20] Suceav˘a, B.D., A Geometric Interpretation of Curvature Inequalities on Hypersurfaces via Ravi Substitutions in the Euclidean Plane. Math. Intelligencer 40 (2018), 50–54.
• [21] Suceav˘a, B.D. and Vajiac, M. B. Remarks on Chen’s fundamental inequality with classical curvature invariants in Riemannian spaces. An. Ştiin¸t. Univ. Al. I. Cuza Ia¸si. Mat. (N.S.) 54 (2008), no. 1, 27–37.