Quadric representation of a submanifold
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- by Ivko Dimitrić PDF
- Proc. Amer. Math. Soc. 114 (1992), 201-210 Request permission
Abstract:
If $x: M^n \to E^m$ is an isometric immersion of a smooth manifold into a Euclidean space then the map $\tilde {x} = x x^{\mathrm {t}}$]> (t denotes transpose) is called the quadric representation of $M$. $\tilde {x}$ is said to be of finite type ($k$-type) if it can be decomposed into a sum of finitely many $(k)$ eigenfunctions of Laplacian from different eigenspaces. We study map $\tilde {x}$ in general, especially as related to the condition of being of finite type. Certain classification results are obtained for manifolds with $1$-and $2$-type quadric representation.References
- Bang-yen Chen, Geometry of submanifolds, Pure and Applied Mathematics, No. 22, Marcel Dekker, Inc., New York, 1973. MR 0353212
- Bang-Yen Chen, Total mean curvature and submanifolds of finite type, Series in Pure Mathematics, vol. 1, World Scientific Publishing Co., Singapore, 1984. MR 749575, DOI 10.1142/0065 I. Dimitrić, Quadric representation and submanifolds of finite type, Thesis, Michigan State Univ., 1989.
- Antonio Ros, Eigenvalue inequalities for minimal submanifolds and $P$-manifolds, Math. Z. 187 (1984), no. 3, 393–404. MR 757479, DOI 10.1007/BF01161955
- Kunio Sakamoto, Planar geodesic immersions, Tohoku Math. J. (2) 29 (1977), no. 1, 25–56. MR 470913, DOI 10.2748/tmj/1178240693
- Tsunero Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380–385. MR 198393, DOI 10.2969/jmsj/01840380
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 114 (1992), 201-210
- MSC: Primary 53C40; Secondary 53C42
- DOI: https://doi.org/10.1090/S0002-9939-1992-1086324-1
- MathSciNet review: 1086324