Conformal motion of contact manifolds with characteristic vector field in the \(k\)-nullity distribution
type: Item
instance of: scholarly article
title: Conformal motion of contact manifolds with characteristic vector field in the \(k\)-nullity distribution
zbMATH Open document ID: 0871.53025
author: Ramesh Sharma
author: David E. Blair
published in: Illinois Journal of Mathematics
publication date: 1997-09-24T00:00:00Z
review text: Let \(M^{2n+1}\) be a contact metric manifold and \((\varphi,\xi,\eta,g)\) its contact structure. The \(k\)-nullity distribution is defined on \(M^{2n+1}\) by \(N(k):p\mapsto N_p(k)=\{Z\in T_pM:R(X,Y)Z=k(g(Y,Z)X-g(X,Z)Y)\) for an \(X,Y\in T_pM\) and a real number \(k\}\). A vector field \(v\) on \(M^{2n+1}\) is an infinitesimal contact transformation if \(L_v\eta=f\eta\) for some function \(f\) where \(L\) denotes the Lie-derivative operator, while \(v\) is an automorphism if \(v\) leaves all the structure tensors \(\varphi\), \(\xi\), \(\eta\), \(g\) invariant. The first result of this paper is the following: (i) Let \(v\) be a non-isometric conformal motion on a 3-dimensional Sasakian manifold \(M\). If the scalar curvature of \(M\) is constant, then \(M\) is of constant curvature 1 and \(v\) is special concircular. If \(v\) is an infinitesimal contact transformation, then \(v\) is special concircular. Hence in either case, if in addition \(M\) is complete and connected, then it is isometric to a unit sphere. This result extends to dimension 3 Okumura's result [see \textit{M. Okumura}, Tôhoku Math. J., II. Ser. 14, 398-412 (1962; Zbl 0107.16201)]. Next the authors prove the following result: Let \(M^{2n+1}\) be a contact metric manifold with \(\xi\) in \(N(k)\) and \(v\) a conformal motion on \(M^{2n+1}\). For \(n>1\), \(M^{2n+1}\) is either Sasakian or \(v\) is Killing. In the second case \(v\) is an automorphism of the associated contact metric structure. For \(n=1\), \(M\) is either flat or Sasakian or \(v\) is an automorphism of the contact metric structure.
reviewed by: Christos Baikoussis
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 1004569
zbMATH Keywords: Killing field
zbMATH Keywords: contact metric manifold
zbMATH Keywords: conformal motion
zbMATH Keywords: Sasakian manifold
zbMATH Keywords: scalar curvature
zbMATH Keywords: concircular
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1004569