A characterization of pseudo-Einstein real hypersurfaces in a quaternionic projective space (Q1375741)

Let \(M\) be a real hypersurface of a quaternionic projective space \(\mathbb{H} P^n\), \(n\geq 3\), endowed with the Fubini-Study metric of constant quaternionic sectional curvature 4. Denote by \(g\) the Riemannian metric, by \(S\) the Ricci tensor, and by \(\rho\) the scalar curvature of \(M\). Suppose there exists a function \(\alpha\) on \(M\) and a canonical local basis of the quaternionic Kähler structure of \(\mathbb{H} P^n\) such that the corresponding structure vector fields \(U_1\), \(U_2\), \(U_3\) satisfy \(g(SU_i,U_i) =\alpha\). Then \(|S|^2 \geq 3\alpha^2 +(\rho-3 \alpha)^2/4(n-1)\), and equality holds if and only if \(M\) is an open part of either a geodesic hypersphere in \(\mathbb{H} P^n\) or of a tube of radius \(r\) over a totally geodesic \(\mathbb{H} P^k\) for some \(k\in\{1, \dots, n-2\}\), where \(0<r<\pi/2\) with \(\text{cot}^2(r)= (4k+2)/(4n-4k-2)\).