Bearing in mind this historical context, in our present project we obtain new relationships involving intrinsic and extrinsic curvature invariants, under natural geometric conditions.
📌 3 Key Takeaways
🎯
Objective & Hypothesis:Nash’s Theorem, any Riemannian manifold can be embedded into a Euclidean ambient space with dimension sufficiently large.
📊
Key Findings & Data: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.
💡
Conclusion & Impact:Bearing in mind this historical context, in our present project we obtain new relationships involving intrinsic and extrinsic curvature invariants, under natural geometric conditions.
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.
Suceava, B., Tran, A. D. (2024). B.-Y. Chen-Type Inequalities for Three Dimensional Smooth Hypersurfaces. International Electronic Journal of Geometry, Volume: 17 Issue: 1. https://doi.org/10.36890/iejg.1366352
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).
Bonnet, O.: Sur quelque propriétés des lignes géodésiques, Comptes rendus de l’Academie des Sciences, 11. 1311–1313 (1855).
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).
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).
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
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
Chen, B.-Y.: Geometry of submanifolds, Marcel Dekker, New York (1973).
Chen, B.-Y.: Geometry of submanifolds and its applications. Science University of Tokyo (1981).
Chen, B.-Y.: Some pinching and classification theorems for minimal submanifolds. Arch. Math. 60 (6), 568–578 (1993). https://doi.org/10.1007/BF01236084
Chen, B.-Y.: Pseudo-Riemannian submanifolds, δ-invariants and Applications. World Scientific (2011).
Chern, S.-S.: Minimal Submanifolds in a Riemannian Manifold. University of Kansas, Department of Mathematics Technical Report 19. Lawrence, Kansas (1968).
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).
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.
Djori´c, M., Okumura, M.: CR Submanifolds of Complex Projective Space. Developments in Mathematics, 19. Springer (2010).
do Carmo, M. P.: Riemannian Geometry. Birkhäuser (1992).
Kobayashi, Sh., Nomizu, K.: Foundations of Differential Geometry, vol.II. John Wiley and Sons (1969).
Nash, Jr., J. F.: The imbedding problem for Riemannian manifolds. Ann. Math. 63 (2), 20–63 (1956).
Pinl, M., Ziller, W.: Minimal hypersurfaces in spaces of constant curvature. J. Differential Geometry 11, 335–343 (1976). DOI: 10.4310/jdg/1214433591
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).
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
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).
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
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).
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.