On Normally Flat Einstein Submanifolds
نویسنده
چکیده
The purpose of this paper is to study the second fundamental form of some submanifolds M in Euclidean spaces E" which have flat normal connection. As such, Theorem gives precise expressions for the (essentially 2) Weingarten maps of all 4-dimensional Emstem submanifolds in E6, which are specialized in Corollary 2 to the Rcciflat submanifolds. The main part ofthis paper deals with fiat submanifolds. In 1919, E. Caftan 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" with flat normal connection in lg is totally cylindrical (for all possible dimensions n and rn).
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