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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Quadric representation of a submanifold

Author: Ivko Dimitrić
Journal: Proc. Amer. Math. Soc. 114 (1992), 201-210
MSC: Primary 53C40; Secondary 53C42
MathSciNet review: 1086324
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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.

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Article copyright: © Copyright 1992 American Mathematical Society