Conformal manifolds and closed Weyl structures with vanishing conformal scalar curvature (Q2783760)

The paper is a continuation and a generalization of the previous paper of the author [ibid. 47, 15-33 (1998; Zbl 0935.53018)]. NEWLINENEWLINENEWLINELet \((M,C)\) be a compact connected conformal manifold of dimension \(\geq 3\). Let \(K(M,C)\) be the set of all closed Weyl structures on \((M,C)\) with vanishing conformal scalar curvature. \(K(M,C)\) carries a natural topology. For a \(g\in C\), let \(P^1(M,g)\) be the real vector space of 1-forms, which are parallel with respect to the Levi-Civita connection of \(g\). Denote \(p(M,C)= \text{ max}\{\text{dim }P^1(M,g)\mid g\in C\;\text{ and } g \text{ has constant scalar curvature}\}\). The main result states that if \(1\leq p(M,C)=n(M,C)\) (\(n(M,C)\) being a certain conformal invariant of integer type), then \(K(M,C)\) is homeomorphic to the Euclidean sphere \(\mathbb{S}^{p(M,C)-1}\). The paper contains also many another interesting results concerning closed Weyl structures, especially, those with vanishing scalar curvature.