Conformally flat immersions and flatness of the normal connection

Maria Helena Noronha

Vol. 138 (1989), No. 1, 145–150

DOI: 10.2140/pjm.1989.138.145

Abstract

B. Y. Chen and T. Teng affirmed that almost umbilic isometric immersions of an n-dimensional manifold in Rⁿ⁺² have zero normal curvature (R^⊥ = 0). In this paper we exhibit a counterexample to this statement and we prove that either R^⊥ = 0 or there exists (locally) an isometric immersion of this manifold in Rⁿ⁺² with R^⊥ = 0. These immersions are conformally flat and we study their local geometry.

Mathematical Subject Classification 2000

Primary: 53B25

Secondary: 53C12

Milestones

Received: 7 January 1987

Published: 1 May 1989