ISOMETRIC IMMERSION OF MINIMAL SPHERICAL SUBMANIFOLD VIA THE SECOND STANDARD IMMERSION OF THE SPHERE
Main Article Content
Abstract
Let $M^n$ be a $n$-dimensional compact connected minimal submanifold of the unit sphere $S^{n+p}(1)$. In this paper we study the isometric immersion of $M^n$ into $SM(n +p + 1)$ via the second standard immersion of $S^{n+p}(1)$. We obtain some integral inequalities m terms of the spectrum of the Laplace operator of $M^n$ and find some restrictions on such immersions.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
References
Barros, M. and Chen, B. Y., "Spherical submanifolds which are of 2-type via the second standard immersion of the sphere," Nagoya Math. J. Vol. 108 (1987), 77-91.
Barros, M. and Urbano, F., "Spectral geometry Of Minimal Surface In the Sphere," Tohoku Math. Journ. 39 (1987), 575-588.
Chavel, I., "Eigenvalues in Riemannian Geometry," Academic Press. 1984.
Chen, B. Y., "Total Mean Curvature and Submanifolds of Finite Type," World Scientific 1984.
Chern, S. S., Do Carmo, M. and Kobayashi, S., "Minimal submanifolds of a sphere with second fundamental form of constant length," Fundamental analysis and related field. Springer-Verlag, 1970.
Dimitric I "Quadric representation of a submanifols" Proc. AMS., Vol. 114, No. 1 (1992) 201-215.
Hsiang, W. H., "Minimal cones and the spherical Bernstein problem I," Ann. of Math., 118 (1983), 61-73. II, Invent. Math. 74 (1983), 351-369.
Ros, A., "Eigenvalue inequalities for minimal submanifolds and P-manifolds," Math. Z. 187 (1984) 393-404.
Simons, J., "Minimal varieties in the Riemannian manifolds," Ann. of Math. 88 (1968) 62-105.
Takahashi, T., "Homogeneous Kaehler submanifolds in complex projective spaces," Japan J. Math. 4 (1978) 171-219.
Tomtier, P., "The spherical Bernstein problem in even dimensions," Bull. AMS., 11.(1984) 183-185.
Wallach N. R "Minimal immersions of symmetric space into sphere, in symmetric spaces," Dekker, 1972, pp. 1-40.