Research Article
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Year 2019, , 57 - 60, 27.03.2019
https://doi.org/10.36890/iejg.545755

Abstract

References

  • [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.

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

Year 2019, , 57 - 60, 27.03.2019
https://doi.org/10.36890/iejg.545755

Abstract

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.

References

  • [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.
There are 21 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

Leonard M. Giugiuc This is me

Bogdan D. Suceava

Publication Date March 27, 2019
Published in Issue Year 2019

Cite

APA Giugiuc, L. M., & Suceava, B. D. (2019). A Characterization of Cylinders and an Estimate for Mean Curvature of Convex Euclidean Hypersurfaces Satisfying a Cylindrical Condition. International Electronic Journal of Geometry, 12(1), 57-60. https://doi.org/10.36890/iejg.545755
AMA Giugiuc LM, Suceava BD. A Characterization of Cylinders and an Estimate for Mean Curvature of Convex Euclidean Hypersurfaces Satisfying a Cylindrical Condition. Int. Electron. J. Geom. March 2019;12(1):57-60. doi:10.36890/iejg.545755
Chicago Giugiuc, Leonard M., and Bogdan D. Suceava. “A Characterization of Cylinders and an Estimate for Mean Curvature of Convex Euclidean Hypersurfaces Satisfying a Cylindrical Condition”. International Electronic Journal of Geometry 12, no. 1 (March 2019): 57-60. https://doi.org/10.36890/iejg.545755.
EndNote Giugiuc LM, Suceava BD (March 1, 2019) A Characterization of Cylinders and an Estimate for Mean Curvature of Convex Euclidean Hypersurfaces Satisfying a Cylindrical Condition. International Electronic Journal of Geometry 12 1 57–60.
IEEE L. M. Giugiuc and B. D. Suceava, “A Characterization of Cylinders and an Estimate for Mean Curvature of Convex Euclidean Hypersurfaces Satisfying a Cylindrical Condition”, Int. Electron. J. Geom., vol. 12, no. 1, pp. 57–60, 2019, doi: 10.36890/iejg.545755.
ISNAD Giugiuc, Leonard M. - Suceava, Bogdan D. “A Characterization of Cylinders and an Estimate for Mean Curvature of Convex Euclidean Hypersurfaces Satisfying a Cylindrical Condition”. International Electronic Journal of Geometry 12/1 (March 2019), 57-60. https://doi.org/10.36890/iejg.545755.
JAMA Giugiuc LM, Suceava BD. A Characterization of Cylinders and an Estimate for Mean Curvature of Convex Euclidean Hypersurfaces Satisfying a Cylindrical Condition. Int. Electron. J. Geom. 2019;12:57–60.
MLA Giugiuc, Leonard M. and Bogdan D. Suceava. “A Characterization of Cylinders and an Estimate for Mean Curvature of Convex Euclidean Hypersurfaces Satisfying a Cylindrical Condition”. International Electronic Journal of Geometry, vol. 12, no. 1, 2019, pp. 57-60, doi:10.36890/iejg.545755.
Vancouver Giugiuc LM, Suceava BD. A Characterization of Cylinders and an Estimate for Mean Curvature of Convex Euclidean Hypersurfaces Satisfying a Cylindrical Condition. Int. Electron. J. Geom. 2019;12(1):57-60.