A Riemannian invariant and its applications to submanifold theory
Let \(M^n\) be a Riemannian manifold with scalar curvature \(r\). If \(K (\pi)\) denotes the sectional curvature of the 2-plane \(\pi\), then \(\delta_M(p) = r(p) - \inf K(p)\) (where the infimum is taken over all the 2-planes in \(p \in M)\) defines a Riemannian invariant [introduced by the author in ``A Riemannian invariant for submanifolds in space forms and its applications'', Geometry and topology of submanifolds, VI (World Scientific), 58-81 (1994; Zbl 0865.53050)]. Here the author uses \(\delta_M\) to derive necessary conditions for the existence of slant immersions. E.g., it is proved: Let \(M\) be compact, with finite \(\pi_1\). i) If \(\delta_M > 0\), \(M\) admits no slant immersion in any flat Kähler manifold; ii) if \(2 \delta_M > (n - 2) (n + 1)c\), \(M\) admits no Lagrangian immersion in \(\mathbb{C} P^n (4c)\). The paper is rich in examples and ends with a list of open problems in the theory of slant immersions.
scientific article; zbMATH DE number 785192
scientific article