Characterizing a class of totally real submanifolds of S⁶ by their sectional curvatures

The first author introduced in a previous paper an important Riemannian invariant for a Riemannian manifold, namely take the scalar curvature function and subtract at each point the smallest sectional curvature at that point. He also proved a sharp inequality for this invariant for submanifolds of real space forms. In this paper we study totally real submanifolds in the nearly Kahler six-sphere that realize the equality in that inequality. In this way we characterize a class of totally real warped product immersions by one equality involving their sectional curvatures.