International Journal of Mathematics and Mathematical Sciences 
Volume 2003 (2003), Issue 11, Pages 711-716
doi:10.1155/S0161171203201137

A note on Chen's basic equality for submanifolds in a Sasakian space form

Mukut Mani Tripathi,1 Jeong-Sik Kim,2 and Seon-Bu Kim3

 1Department of Mathematics and Astronomy, Lucknow University, Lucknow 226 007, India
2Department of Mathematics Education, Sunchon National University, Sunchon 540-742, Korea
3Department of Mathematics, Chonnam National University, Kwangju 500-757, Korea
Received 28 January 2002

Abstract

It is proved that a Riemannian manifold M isometrically immersed in a Sasakian space form M˜(c) of constant φ-sectional curvature c<1, with the structure vector field ξ tangent to M, satisfies Chen's basic equality if and only if it is a 3-dimensional minimal invariant submanifold.