Reference Ic/88/392
REFERENCE IC/88/392 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS THE LOCAL STRUCTURE OF A 2-CODIMENSIONAL CONFORMALLY FLAT SUBMANIFOLD IN AN EUCLIDEAN SPACE lRn+2 G. Zafindratafa INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION IC/88/392 ABSTRACT International Atomic Energy Agency and Since 1917, many mathematicians studied conformally flat submanifolds (e.g. E. Cartan, B.Y. Chen, J.M. Morvan, L. Verstraelen, M. do Carmo, N. Kuiper, etc.). United Nations Educational Scientific and Cultural Organization In 1917, E. Cartan showed with his own method that the second fundamental form INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS of a conforraally flat hypersurface of IR™+1 admits an eigenvalue of multiplicity > n — 1. In order to generalize such a result, in 1972, B.Y. Chen introduced the notion of quasiumbilical submanifold. A submanifold M of codimensjon JV in 1R"+W is quasiumbilical if and only if there exists, locally at each point of M, an orthonormal frame field {d,...,£jv} of the normal space so that the Weingarten tensor of each £„ possesses an eigenvalue of multiplicity > n — 1. THE LOCAL STRUCTURE OF A 2-CODIMENSIONAL CONFORMALLY FLAT SUBMANIFOLD The first purpose of this work is to resolve the following first problem: IN AN EUCLIDEAN SPACE JRn+;1 * Is there any equivalent definition of quasiumbilictty which does not use either any local frame field on the normal bundle or any particular sections on the tangent bundle? B.Y. Chen and K. Yano proved in 1972 that a quasiumbilical submanifold of G. Zafmdratafa, ** codimension N > 1 in JR"+" is conformally flat; moreover in the case of codimension International Centre for Theoretical Physics, Trieste, Italy.
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