International Journal of Mathematics and Mathematical Sciences
Volume 20 (1997), Issue 3, Pages 497-501
On normally flat Einstein submanifolds
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
Copyright © 1997 Leopold Verstraelen and Georges Zafindratafa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The purpose of this paper is to study the second fundamental form of some submanifolds
in Euclidean spaces which have flat normal connection. As such, Theorem gives precise
expressions for the (essentially 2) Weingarten maps of all 4-dimensional Einstein submanifolds in ,
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 in a
Euclidean space is totally cylindrical. Moreover, he asserted without proof the existence of flat nontotally
cylindrical submanifolds of dimension in Euclidean spaces. We will comment on this
assertion, and in this respect will prove, in Theorem 3, that every flat submanifold with flat normal
connection in is totally cylindrical (for all possible dimensions and ).