J Geom Anal DOI 10.1007/s12220-013-9418-5 Entire Conformal Killing Graphs in Foliated Riemannian Spaces Henrique F. de Lima · Joseilson R. de Lima · Marco Antonio L. Velásquez Received: 23 October 2012 © Mathematica Josephina, Inc. 2013 Abstract We study the geometry of entire conformal Killing graphs, that is, graphs constructed through the flow generated by a complete conformal Killing vector field V and which are defined over an integral leaf of the foliation V ⊥ orthogonal to V . In this setting, under a suitable restriction on the norm of the gradient of the function z which determines such a graph Σ(z), we establish sufficient conditions to ensure that Σ(z) is totally umbilical and, in particular, an integral leaf of V ⊥ . Keywords Conformal Killing vector fields · Conformal Killing graphs · Totally umbilical hypersurfaces · r th mean curvatures · Newton transformations Mathematics Subject Classification 53C42 · 53C12 1 Introduction Conformal Killing vector fields are important objects which have been widely used in order to understand the geometry of submanifolds and, more particularly, of hy- persurfaces immersed in Riemannian spaces. In this setting, Montiel [17] has studied The first author is partially supported by CNPq, Brazil. The first and third authors are partially supported by CAPES/CNPq, Brazil, grant Casadinho/Procad 552.464/2011-2. The authors would like to thank the referee for giving some valuable suggestions which improved the paper. H.F. de Lima ( ) · J.R. de Lima · M.A.L. Velásquez Departamento de Matemática e Estatística, Universidade Federal de Campina Grande, 58109-970 Campina Grande, Paraíba, Brazil e-mail: henrique@dme.ufcg.edu.br J.R. de Lima e-mail: joseilson@dme.ufcg.edu.br M.A.L. Velásquez e-mail: marcolazarovelasquez@gmail.com