Pinching theorems for C-totally real submanifolds of Sasakian space forms
A C-totally real submanifold of a Sasakian manifold \(\tilde M^{2n+1}\) is an integral submanifold of the contact subbundle of \(\tilde M^{2n+1}\); such submanifolds have dimension at most \(n\). A Sasakian space form, \(\tilde M^{2n+1}(c)\), is a Sasakian manifold of constant \(\phi\)-sectional curvature, \(c\); this is analogous to the notion of a Kähler manifold of constant holomorphic curvature. For a reference to Sasakian manifolds and these ideas, see the paper or the reviewer's monograph [Contact manifolds in Riemannian geometry. Lect. Notes Math. 509. Berlin etc.: Springer-Verlag (1976; Zbl 0319.53026)] The main results of the paper are the following: If M is an n-dimensional compact, minimal, C-totally real submanifold of \(\tilde M^{2n+1}(c)\) and if all sectional curvatures \(K\) of \(M\) satisfy \(K\geq 0\), then M has constant scalar curvature. Moreover if \(n=2\), then the Gaussian curvature \(K\) of \(M\) is either 0 or \((c+3)/4\) and an example is given. These results are stronger than previously known ones due to the method of proof. The method is to use the integral formulas of \textit{A. Ros} [Ann. Math. (2) 121, 377--382 (1983; Zbl 0589.53065)] on the unit tangent bundle of a compact Riemannian manifold.
scientific article; zbMATH DE number 4067812