International Journal of Mathematics and Mathematical Sciences 
Volume 20 (1997), Issue 3, Pages 497-501
doi:10.1155/S0161171297000677

On normally flat Einstein submanifolds

Leopold Verstraelen1 and Georges Zafindratafa2

 1K.U. Leuven, Department of Mathematics, Celestijnenlaan 200B, Leuven 3001, Belgium
2Université de Valenciennes, Institut des Sciences et Techniques, B.P. 311, Cedex, Valencienes F-59304, France
Received 21 December 1990; Revised 2 April 1993

Abstract

The purpose of this paper is to study the second fundamental form of some submanifolds Mn in Euclidean spaces 𝔼m which have flat normal connection. As such, Theorem gives precise expressions for the (essentially 2) Weingarten maps of all 4-dimensional Einstein submanifolds in 𝔼6, 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 Mn with flat normal connection in 𝔼m is totally cylindrical (for all possible dimensions n and m).