Compact Einstein-Weyl manifolds and the associated constant
Let \(M\) be a conformal manifold with a conformal structure \([g]\). A torsion-free connection \(D\) on \(M\) is called a Weyl structure if \(Dg=\omega\otimes g\) for a 1-form \(\omega\). A Weyl structure \((D,[g])\) on \(M\) is Einstein-Weyl if \(\text{Ric}^D(X,Y)+ \text{Ric}^D(Y,X)=\lambda g(X,Y)\), where \(\lambda\) is a function on \(M\). For each compact Einstein-Weyl manifold \(M\) with coclosed 1-form \(\omega\) the scalar \(c=s^D+\tfrac{n(n-4)}4| \omega| ^2\) is constant, where \(s^D\) is the conformal scalar curvature, that is, the metric trace of \(\text{Ric}^D\). The constant \(c\) is called an associated constant. In this paper, the author characterizes compact Einstein-Weyl manifolds in terms of the sign of the associated constant \(c\). Among other results, it is shown that if \(c>0\), then \(\text{Ric}_g\geq 0\) and the first Betti number \(b_1(M) \leq 1\). And if \(b_1(M)=1\), then the universal covering \(\widetilde M\) of \(M\) is \(N\times{\mathbb R}\), where \(N\) is an Einstein manifold of positive scalar curvature.
scientific article; zbMATH DE number 1216672