On normally flat Einstein submanifolds

Leopold Verstraelen and Georges Zafindratafa

International Journal of Mathematics and Mathematical Sciences, 1997, vol. 20, 1-5

Abstract:

The purpose of this paper is to study the second fundamental form of some submanifolds M n in Euclidean spaces 𝔼 m which have flat normal connection . As such, Theorem gives precise expressions for the (essentially 2) Weingarten maps of all 4-dimensional Einstein submanifolds in 𝔼 6 , which are specialized in Corollary 2 to the Ricci flat submanifolds. The main part of this paper deals with flat submanifolds. In 1919, E. Cartan proved that every flat submanifold of dimension ≤ 3 in a Euclidean space is totally cylindrical. Moreover, he asserted without proof the existence of flat nontotally cylindrical submanifolds of dimension > 3 in Euclidean spaces. We will comment on this assertion, and in this respect will prove, in Theorem 3, that every flat submanifold M n with flat normal connection in 𝔼 m is totally cylindrical (for all possible dimensions n and m ).

Date: 1997
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:927596

DOI: 10.1155/S0161171297000677

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