Conformal motion of contact manifolds with characteristic vector field in the \(k\)-nullity distribution
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.
scientific article; zbMATH DE number 1004569