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B.-Y. Chen-Type Inequalities for Three Dimensional Smooth Hypersurfaces

Year 2024, Volume: 17 Issue: 1, 146 - 152, 23.04.2024
https://doi.org/10.36890/iejg.1366352

Abstract

By J.F. Nash’s Theorem, any Riemannian manifold can be embedded into a Euclidean ambient
space with dimension sufficiently large. S.-S. Chern pointed out in 1968 that a key technical
element in applying Nash’s Theorem effectively is finding useful relationships between intrinsic
and extrinsic elements that are characterizing immersions. After 1993, when a groundbreaking
work written by B.-Y.Chen on this theme was published, many explorations pursued this
important avenue. Bearing in mind this historical context, in our present project we obtain
new relationships involving intrinsic and extrinsic curvature invariants, under natural geometric
conditions.

References

  • [1] Andreescu, T., St˘anean, M.: New, Newer, and Newest Inequalities. XYZ Press (2021).
  • [2] Barbosa, L., do Carmo, M., A necessary condition for a metric in Mn to be minimally immersed in Rn+1. An. Acad. Bras. Cienc. 50, 445–454 (1978).
  • [3] Bonnet, O.: Sur quelque propriétés des lignes géodésiques, Comptes rendus de l’Academie des Sciences, 11. 1311–1313 (1855).
  • [4] Brubaker, N. D., Suceava, B. D.: A geometric interpretation of Cauchy-Schwarz inequality in terms of Casorati curvature, International Electronic Journal of Geometry 11 (1), 48–51 (2018).
  • [5] Brzycki, B., Giesler, M. D. , Gomez, K., Odom, L. H. , Suceav˘a, B.D.: A ladder of curvatures for hypersurfaces in the Euclidean ambient space. Houston Journal of Mathematics. 40 (4), 1347–1356 (2014).
  • [6] Cartan, É.: La déformation des hypersurfaces dans l’espace eucliden réel a n dimensions, Bull. Soc. Math. France, 44, 65–99 (1916). DOI: 10.24033/bsmf.964
  • [7] Casorati, F.: Mesure de la courbure des surfaces suivant l’idée commune. Ses rapports avec les mesures de courbure gaussienne et moyenne. Acta Math. 14 (1), 95–110 (1890). DOI: 10.1007/BF02413317
  • [8] Chen, B.-Y.: Geometry of submanifolds, Marcel Dekker, New York (1973).
  • [9] Chen, B.-Y.: Geometry of submanifolds and its applications. Science University of Tokyo (1981).
  • [10] Chen, B.-Y.: Some pinching and classification theorems for minimal submanifolds. Arch. Math. 60 (6), 568–578 (1993). https://doi.org/10.1007/BF01236084
  • [11] Chen, B.-Y.: Pseudo-Riemannian submanifolds, δ-invariants and Applications. World Scientific (2011).
  • [12] Chern, S.-S.: Minimal Submanifolds in a Riemannian Manifold. University of Kansas, Department of Mathematics Technical Report 19. Lawrence, Kansas (1968).
  • [13] Chern, S.-S., Osserman, R.: Remarks on the Riemannian metric of a minimal submanifold, Lecture Notes in Math., 894, 49–90. Springer-Verlag. Berlin-New York (1981).
  • [14] Conley, C.T.R. , Etnyre, R., Gardener, B., Odom, L.H., Suceav˘a, B. D., New curvature inequalities for hypersurfaces in the Euclidean ambient space, Taiwanese J. Math. 17 (3) 885–895 (2013). DOI: 10.11650/tjm.17.2013.2504.
  • [15] Djori´c, M., Okumura, M.: CR Submanifolds of Complex Projective Space. Developments in Mathematics, 19. Springer (2010).
  • [16] do Carmo, M. P.: Riemannian Geometry. Birkhäuser (1992).
  • [17] Kobayashi, Sh., Nomizu, K.: Foundations of Differential Geometry, vol.II. John Wiley and Sons (1969).
  • [18] Myers, S. B.: Riemmannian manifolds with positive curvature. Duke Math. J. 8, 401–404 (1941). DOI: 10.1215/S0012-7094-41-00832-3
  • [19] Nash, Jr., J. F.: The imbedding problem for Riemannian manifolds. Ann. Math. 63 (2), 20–63 (1956).
  • [20] Pinl, M., Ziller, W.: Minimal hypersurfaces in spaces of constant curvature. J. Differential Geometry 11, 335–343 (1976). DOI: 10.4310/jdg/1214433591
  • [21] Suceava, B. D.: Fundamental inequalities and strongly minimal submanifolds, Recent Advances in Riemannian and Lorentzian Geometries, eds.K. L. Duggal, K.L., Sharma, R. Contemporary Mathematics vol. 337. American Mathematical Society. 155–170 (2003).
  • [22] Suceava, B. D.: The amalgamatic curvature and the orthocurvatures of three dimensional hypersurfaces in the Euclidean space. Publ. Math. Debrecen. 87, 35–46 (2015). DOI: 10.5486/PMD.2015.7003
  • [23] Suceava, B. D., Vajiac, M. B.: Remarks on Chen’s fundamental inequality with classical curvature invariants in Riemannian spaces. An. Stiint. Univ. Al. I. Cuza Ia¸si. Mat.(NS) 54, 27–37 (2008).
  • [24] Vîlcu, G. E., Curvature Inequalities for Slant Submanifolds in Pointwise Kenmotsu Space Forms. In: Chen, B.-Y., Shahid, M.H., Al-Solamy, F. (eds) Contact Geometry of Slant Submanifolds. Springer, Singapore, 2022. DOI: 10.1007/978-981-16-0017-3_2
Year 2024, Volume: 17 Issue: 1, 146 - 152, 23.04.2024
https://doi.org/10.36890/iejg.1366352

Abstract

References

  • [1] Andreescu, T., St˘anean, M.: New, Newer, and Newest Inequalities. XYZ Press (2021).
  • [2] Barbosa, L., do Carmo, M., A necessary condition for a metric in Mn to be minimally immersed in Rn+1. An. Acad. Bras. Cienc. 50, 445–454 (1978).
  • [3] Bonnet, O.: Sur quelque propriétés des lignes géodésiques, Comptes rendus de l’Academie des Sciences, 11. 1311–1313 (1855).
  • [4] Brubaker, N. D., Suceava, B. D.: A geometric interpretation of Cauchy-Schwarz inequality in terms of Casorati curvature, International Electronic Journal of Geometry 11 (1), 48–51 (2018).
  • [5] Brzycki, B., Giesler, M. D. , Gomez, K., Odom, L. H. , Suceav˘a, B.D.: A ladder of curvatures for hypersurfaces in the Euclidean ambient space. Houston Journal of Mathematics. 40 (4), 1347–1356 (2014).
  • [6] Cartan, É.: La déformation des hypersurfaces dans l’espace eucliden réel a n dimensions, Bull. Soc. Math. France, 44, 65–99 (1916). DOI: 10.24033/bsmf.964
  • [7] Casorati, F.: Mesure de la courbure des surfaces suivant l’idée commune. Ses rapports avec les mesures de courbure gaussienne et moyenne. Acta Math. 14 (1), 95–110 (1890). DOI: 10.1007/BF02413317
  • [8] Chen, B.-Y.: Geometry of submanifolds, Marcel Dekker, New York (1973).
  • [9] Chen, B.-Y.: Geometry of submanifolds and its applications. Science University of Tokyo (1981).
  • [10] Chen, B.-Y.: Some pinching and classification theorems for minimal submanifolds. Arch. Math. 60 (6), 568–578 (1993). https://doi.org/10.1007/BF01236084
  • [11] Chen, B.-Y.: Pseudo-Riemannian submanifolds, δ-invariants and Applications. World Scientific (2011).
  • [12] Chern, S.-S.: Minimal Submanifolds in a Riemannian Manifold. University of Kansas, Department of Mathematics Technical Report 19. Lawrence, Kansas (1968).
  • [13] Chern, S.-S., Osserman, R.: Remarks on the Riemannian metric of a minimal submanifold, Lecture Notes in Math., 894, 49–90. Springer-Verlag. Berlin-New York (1981).
  • [14] Conley, C.T.R. , Etnyre, R., Gardener, B., Odom, L.H., Suceav˘a, B. D., New curvature inequalities for hypersurfaces in the Euclidean ambient space, Taiwanese J. Math. 17 (3) 885–895 (2013). DOI: 10.11650/tjm.17.2013.2504.
  • [15] Djori´c, M., Okumura, M.: CR Submanifolds of Complex Projective Space. Developments in Mathematics, 19. Springer (2010).
  • [16] do Carmo, M. P.: Riemannian Geometry. Birkhäuser (1992).
  • [17] Kobayashi, Sh., Nomizu, K.: Foundations of Differential Geometry, vol.II. John Wiley and Sons (1969).
  • [18] Myers, S. B.: Riemmannian manifolds with positive curvature. Duke Math. J. 8, 401–404 (1941). DOI: 10.1215/S0012-7094-41-00832-3
  • [19] Nash, Jr., J. F.: The imbedding problem for Riemannian manifolds. Ann. Math. 63 (2), 20–63 (1956).
  • [20] Pinl, M., Ziller, W.: Minimal hypersurfaces in spaces of constant curvature. J. Differential Geometry 11, 335–343 (1976). DOI: 10.4310/jdg/1214433591
  • [21] Suceava, B. D.: Fundamental inequalities and strongly minimal submanifolds, Recent Advances in Riemannian and Lorentzian Geometries, eds.K. L. Duggal, K.L., Sharma, R. Contemporary Mathematics vol. 337. American Mathematical Society. 155–170 (2003).
  • [22] Suceava, B. D.: The amalgamatic curvature and the orthocurvatures of three dimensional hypersurfaces in the Euclidean space. Publ. Math. Debrecen. 87, 35–46 (2015). DOI: 10.5486/PMD.2015.7003
  • [23] Suceava, B. D., Vajiac, M. B.: Remarks on Chen’s fundamental inequality with classical curvature invariants in Riemannian spaces. An. Stiint. Univ. Al. I. Cuza Ia¸si. Mat.(NS) 54, 27–37 (2008).
  • [24] Vîlcu, G. E., Curvature Inequalities for Slant Submanifolds in Pointwise Kenmotsu Space Forms. In: Chen, B.-Y., Shahid, M.H., Al-Solamy, F. (eds) Contact Geometry of Slant Submanifolds. Springer, Singapore, 2022. DOI: 10.1007/978-981-16-0017-3_2
There are 24 citations in total.

Details

Primary Language English
Subjects Algebraic and Differential Geometry
Journal Section Research Article
Authors

Bogdan Suceava 0000-0003-3361-3201

Anh Du Tran 0009-0006-3207-7901

Early Pub Date April 6, 2024
Publication Date April 23, 2024
Acceptance Date December 27, 2023
Published in Issue Year 2024 Volume: 17 Issue: 1

Cite

APA Suceava, B., & Tran, A. D. (2024). B.-Y. Chen-Type Inequalities for Three Dimensional Smooth Hypersurfaces. International Electronic Journal of Geometry, 17(1), 146-152. https://doi.org/10.36890/iejg.1366352
AMA Suceava B, Tran AD. B.-Y. Chen-Type Inequalities for Three Dimensional Smooth Hypersurfaces. Int. Electron. J. Geom. April 2024;17(1):146-152. doi:10.36890/iejg.1366352
Chicago Suceava, Bogdan, and Anh Du Tran. “B.-Y. Chen-Type Inequalities for Three Dimensional Smooth Hypersurfaces”. International Electronic Journal of Geometry 17, no. 1 (April 2024): 146-52. https://doi.org/10.36890/iejg.1366352.
EndNote Suceava B, Tran AD (April 1, 2024) B.-Y. Chen-Type Inequalities for Three Dimensional Smooth Hypersurfaces. International Electronic Journal of Geometry 17 1 146–152.
IEEE B. Suceava and A. D. Tran, “B.-Y. Chen-Type Inequalities for Three Dimensional Smooth Hypersurfaces”, Int. Electron. J. Geom., vol. 17, no. 1, pp. 146–152, 2024, doi: 10.36890/iejg.1366352.
ISNAD Suceava, Bogdan - Tran, Anh Du. “B.-Y. Chen-Type Inequalities for Three Dimensional Smooth Hypersurfaces”. International Electronic Journal of Geometry 17/1 (April 2024), 146-152. https://doi.org/10.36890/iejg.1366352.
JAMA Suceava B, Tran AD. B.-Y. Chen-Type Inequalities for Three Dimensional Smooth Hypersurfaces. Int. Electron. J. Geom. 2024;17:146–152.
MLA Suceava, Bogdan and Anh Du Tran. “B.-Y. Chen-Type Inequalities for Three Dimensional Smooth Hypersurfaces”. International Electronic Journal of Geometry, vol. 17, no. 1, 2024, pp. 146-52, doi:10.36890/iejg.1366352.
Vancouver Suceava B, Tran AD. B.-Y. Chen-Type Inequalities for Three Dimensional Smooth Hypersurfaces. Int. Electron. J. Geom. 2024;17(1):146-52.