Submanifolds of constant scalar curvature in a hyperbolic space form (Q1301996)

The main result of this article is: Let \(M^n\) be a closed submanifold of the hyperbolic space \(H^{n+p}(-1)\) with parallel normalized mean curvature vector field \(\xi/H\), where \(\xi\) denotes the mean curvature vector field, and \(H\) the length of \(\xi\). Suppose that the normalized scalar curvature \(R\) is constant and is greater than or equal to \((3n-r)/(3n-3)\). If \(H\) satisfies a certain inequality then \(R\) is positive and \(M\) is either a geodesic distance sphere in \(H^{n+p}(-1)\) or \(M^2\) is a Veronese surface in a totally geodesic sphere \(S^4\) of a geodesic distance sphere \(S^5\) in \(H^{2+p}(-1)\). The inequality involving \(H\) that implies this result also involves the dimension of \(M\) and the second fundamental form.