Submanifolds of Weyl flat manifolds (Q1849371)

A Weyl structure on a manifold \(N\) with conformal class \([g]\) is a torsion-free connection \(\nabla^{[g]}\) such that parallel translation is a conformal transformation. Usually, the Ricci tensor of a Weyl structure is not symmetric. If the symmetric part of the Ricci curvature of \(\nabla^{[g]}\) is proportional to the metrics of the conformal class \([g]\), then \((N,[g],\nabla^{[g]})\) is called an Einstein-Weyl manifold. Let \(i:M\to N\) be an immersed submanifold of \(N\). Let \(h=i^*g\) and denote by \(\pi\) the orthogonal projection from \(i^*TN\) to \(TM\). Let \(\nabla^{[h]}_X Y =\pi(\nabla^{[g]}_X Y)\). Then we can call \((M,[h],\nabla^{[h]})\) a Weyl submanifold of \(N\). Theorem 1. Let \((N,[g],\nabla^{[g]})\) be an \(n\)-dimensional Weyl manifold with closed, non-exact Weyl flat structure and let \((M,[h],\nabla^{[h]})\) be a compact \(m\)-dimensional, \(m\geq 3\), Weyl submanifold with a closed, non-exact Einstein-Weyl structure. Then \(M\) is totally umbilical if and only if it is Weyl totally geodesic. Theorem 2. Let \((N,[g],\nabla^{[g]})\) be an \((m+1)\)-dimensional Weyl manifold with closed, non-exact Weyl flat structure and let \((M,[h],\nabla^{[h]})\) be a compact hypersurface of \(N\) with a closed, non-exact Einstein-Weyl structure. Then one of the following mutually exclusive situations occurs: \text{ (1)} \((M,[h],\nabla^{[h]})\) is Weyl totally geodesic; \text{ (2)} \(M\) has two principal curvatures, one equal to \(\mu\) of multiplicity \(m-1\), and the second one, \(\lambda\), of multiplicity \(1\). In this case, the two complementary orthogonal distributions \(\mathcal{D}_\mu\) and \(\mathcal{D}_\lambda\) defined on \(M\) by the eigenspaces of \(\mu\) and \(\lambda\) are integrable and parallel. Hence \(M\) is locally the Riemannian product of the totally geodesic leaves \(F_\mu\) and \(F_\lambda\) of \(\mathcal{D}_\mu\) and \(\mathcal{D}_\lambda\) respectively. Recently, some examples of Einstein-Weyl structures on almost contact manifolds have been constructed [\textit{P. Matzeu}, Classical Quantum Gravity 17, 5079-5087 (2000; Zbl 0972.83060)], \textit{F. Narita}, Tsukuba J. Math. 22, 87-98 (1998; Zbl 0995.53035]). The last part of the paper is devoted to the analysis of Weyl submanifolds of some special, non compact Weyl manifolds, the \(PC\)-manifolds, introduced in [\textit{J. C. Marrero}, Rend. Mat. Appl. 12, 305-327 (1992; Zbl 0753.53022)] as examples of spaces carrying locally conformal cosymplectic structures.