A classification of contact metric 3-manifolds with constant \(\xi\)-sectional and \(\phi\)-sectional curvatures (Q1610972)

In this paper the authors study \(3\)-dimensional contact metric manifolds with constant \(\xi\)-sectional curvature and \(\varphi\)-sectional curvature or constant norm of the Ricci operator. More precisely, let \(M(\varphi, \xi , \eta , g)\) be a 3-dimensional contact metric manifold and \(l, h\) the operators defined by \(l:=R(., \xi)\xi \) and \( h:={1\over 2}L_{\xi }\varphi ,\) where \(R, L\) are the curvature tensor and the Lie derivative, respectively. Among others, the authors prove the following two main results: 1) Let \(M\) be a \(3\)-dimensional contact metric manifold with constant \(\xi \)-sectional curvature. If the norm of the Ricci operator \(Q\) is constant along \(\xi \), then \(Q\varphi =\varphi Q\) or \(l=0\) with constant scalar curvature and \(\eta (QX)=0\) for all eigenvectors \(X\in \text{ker }(\eta)\) of \(h\) with eigenvalue \(1.\) 2) Let \(M\) be a \(3\)-dimensional contact metric manifold with constant \(\xi \)-sectional curvature \( k \) and constant \(\varphi \)-sectional curvature \(m.\) Then, one of the following conditions holds: (i) \(M\) is Sasakian, (ii) \(Q\varphi =\varphi Q\) and \(m=-k,\) (iii) \(l=0,\) (iv) \(k+m={2\over 3},\) (v) \(k+m=-2.\)