Hypersurfaces with \((f,g,u,v,\lambda)\)-structure of an affinely cosymplectic manifold (Q1824213)

Let \((\phi,E,\eta)\) be a contact structure on a \((2n+1)\)-dimensional manifold \(M\), and let \(i: P\to M\) be an imbedding of a \(2n\)-dimensional manifold \(P\) into \(M\). In this paper, the author shows that if \(E\) and \(N\) are distinct normals to \(P\) then \(P\) has a structure \(f\) satisfying \(f^4+(1+\lambda \eta (N))f^2 + \lambda \eta(N) I = 0\), i.e. a quartic structure, and if \(\lambda =\eta(N)\) then \(f\) becomes \((f,U,V,u,v,\lambda)\)-structure on \(P\) [\textit{K. Yano} and \textit{M. Okumura}, Kōdai Math. Semin. Rep. 22, 401--423 (1970; Zbl 0204.54801)]. Furthermore, some properties of hypersurfaces of cosymplectic manifolds are studied, e.g., if \(P\) is a non-invariant \((f,g,u,v,\lambda)\)-hypersurface of a cosymplectic manifold, then it is totally geodesic if \(f\) is parallel with respect to induced connection.