Characterizing a class of totally real submanifolds of \(S^ 6\) by their sectional curvatures
The first author introduced in a previous paper [Arch. Math. 60, No. 6, 568-578 (1993; Zbl 0811.53060)] an important Riemannian invariant \(\delta_M\) 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. For 3-dimensional totally real submanifolds of the nearly Kähler 6-sphere, this inequality reduces to \(\delta_M \leq 2\). In this paper we study totally real submanifolds in the nearly Kähler six-sphere that realize the equality in that inequality. Such submanifolds have an index of relative nullity at least 1 at every point. Here we put as extra conditions that the relative nullity determines a 1-dimensional distribution, and that the orthogonal complement is involutive. We relate submanifolds satisfying the above conditions to minimal (non-totally geodesic) totally real immersions of surfaces \(M^2\) into \(S^6(1)\) whose ellipse of curvature is a circle. Such surfaces are always linearly full in some totally geodesic \(S^5\). Let \(N\) be a unit vector perpendicular to this \(S^5\). Then \[ x : \left( -{\pi \over 2}, {\pi \over 2}\right) \times M^2 \to S^6(1),\;(t,p) \mapsto \sin (t) N + \cos(t)p \] is a totally real immersion which satisfies the above conditions. Conversely, every totally real (non-totally geodesic) immersion of \(M^3\) into \(S^6(1)\) satisfying the above conditions can be locally obtained in this way. We note that the immersion \(x\) is a warped product immersion in the sense of S. Nölker.
scientific article; zbMATH DE number 794313
scientific article