Eta-Einstein condition on twistor spaces of odd-dimensional Riemannian manifolds (Q884687)

The author defines the twistor space of an odd-dimensional Riemannian manifold \((M,g)\) as the bundle \(\mathcal{C}\) over \(M\) whose fibre at a point \(p\in M\) consists of all pairs \((\varphi, \xi)\) of a skew-symmetric endomorphism \(\varphi\) and a unit vector \(\xi\) of \(T_{p}M\) such that \((\varphi, \xi, g_{p})\) satisfies the algebraic identities in the definition of an almost contact metric structure. The smooth manifold \(\mathcal{C}\) admits two natural partially complex structures \(\Phi_{1}\) and \(\Phi_{2}\), a 1-parameter family of Riemannian metrics \(h_{t}, t>0\) compatible with \(\Phi_{1}\) and \(\Phi_{2}\) and a globally defined \(h_{t}\)-unit vector field \(\chi\) such that \((\Phi_{\alpha}, \chi, h_{t}), \alpha=1,2\) is an almost contact metric structure on \(\mathcal{C}\). The metrics \(h_{t}\) on \(\mathcal{C}\) are never Einstein. A generalization of the Einstein condition adapted to the class of almost contact metric manifolds is the so-called eta-Einstein condition. An almost contact metric structure with contact form \(\eta\) and associated metric \(h\) on an odd-dimensional manifold \(N\) is said to be eta-Einstein if there exist smooth functions \(a\) and \(b\) on \(N\) such that \(\text{Ric}_{h}(X,Y)=ah(X,Y)+b\eta (X) \eta(Y)\), for \(X,Y \in TN\). In the general case, the functions \(a\) and \(b\) are not constant (unlike the Einstein case). The author finds the conditions on an odd-dimensional Riemannian manifold under which its twistor space is eta-Einstein. This can be used to yield an Einstein metric on the tangent bundle of any 3-dimensional manifold of positive constant curvature. The main result of paper is: Let \((M,g)\) be a Riemannian manifold of odd dimension \(n\geq 3\). Then its twistor space \(\mathcal{C}\) endowed with the metric \(h_{t}\) and the contact form \(\eta_{t}=h_{t}(\cdot, \chi)\) is eta-Einstein if and only if \(n=3\), the manifold \((M, g)\) is of positive constant curvature \(\nu\) and \(t\nu =\frac{1}{2}\); in this case we have \(a=\frac{3\nu}{2}\) and \(b=-\frac{\nu}{2}\), for the functions \(a\) and \(b\) on \(\mathcal{C}\) of the expression for \(\text{Ric}_{h}\). It is also proved that if \(M\) is a 3-dimensional manifold of positive constant curvature \(\nu\), a suitable deformation of the Tano type of the metric \(h_{\frac{1}{\nu}}\) on the tangent sphere bundle \(T_{1}M\) yields an Einstein metric on it.