ILLINOIS JOURNAL OF MATHEMATICS Volume 40, Number 4, Winter 1996 CONFORMAL MOTION OF CONTACT MANIFOLDS WITH CHARACTERISTIC VECTOR FIELD IN THE k-NULLITY DISTRIBUTION RAMESH SHARMA AND DAVID E. BLAIR Dedicated to the memory of Professor Kentaro Yano 1. Introduction It is known (see for example, 17]) that if an m-dimensional Riemannian manifold admits a maximal, i.e., an (m + 1)(m + 2)/2-parameter group of conformal motions, then it is conformally flat. It is also known [9] that a conformally flat Sasakian (normal contact metric) manifold is of constant curvature 1. This shows that the existence of maximal conformal group places a severe restriction on the Sasakian manifold. Thus one is led to examine the effect of the existence of a single 1-parameter group of conformal motions on a Sasakian manifold. All the transformations considered in this paper are infinitesimal. Okumura 10] proved that a non-isometric conformal motion of a Sasakian manifold M of dimension 2n + (n > 1) is special concircular and hence if, in addition, M is complete and connected then it is isometric to a unit sphere. The proof is based on Obata’s theorem [8]: "Let M be a complete connected Riemannian manifold of dimension m > 1. In order for M to admit a non-trivial solution p of the system of partial differential equations VVp -c2pg (c a constant > 0), it is necessary and sufficient that M be isometric to a unit sphere of radius 1/c." The purpose of this paper is (i) to extend Okumura’s result to dimension 3 and (ii) to study conformal motion of the more general class of contact metric manifolds (M, r/, s e, p, g) satisfying the condition that the characteristic vector field belongs to the k-nullity distribution N(k): p --> Np(k) {Z in TpM: R(X, Y)Z k(g(Y, Z)X-g(X, Z)Y) for any X, Y in TpM and a real number k} (see Tanno [15]). For k 1, M is Sasakian. For k 0, M is flat in dimension 3 and in dimension 2n + > 3, it is locally the Riemannian product E n+l S" (4) (see Blair [3]). We say that a vector field v on M is an infinitesimal contact transformation 12] if for some function f where denotes the Lie-derivative operator. We also say that a vector field v on M is an automorphism of the contact metric structure if v leaves all the structure tensors r/, , p, g invariant (see [13]). Received May 5, 1995 1991 Mathematics Subject Classification. Primary 53C25; Secondary 53C 15. Research of R. Sharma supported in part by a University of New Haven Faculty Fellowship. (C) 1996 by the Board of Trustees of the University of Illinois Manufactured in the United States of America 553