On $(k,\varepsilon)$-saddle submanifolds of Riemannian manifolds

Abstract

The paper continues our (in collaboration with A. Borisenko [J. Differential Geom. Appl. 20 p., to appear]) discovery of the new classes of $(k,\varepsilon)$-saddle, $(k,\varepsilon)$-parabolic, and $(k,\varepsilon)$-convex submanifolds ( $\,\varepsilon\ge0$). These are defined in terms of the eigenvalues of the 2nd fundamental forms of each unit normal of the submanifold, extending the notion of k-saddle, k-parabolic, k-convex submanifolds ( $\,\varepsilon=0$). It follows that the definition of $(k,\varepsilon)$-saddle submanifolds is equivalent to the existence of $\varepsilon$-asymptotic subspaces in the tangent space. We prove non-embedding theorems of $(k,\varepsilon)$-saddle submanifolds, theorems about 1-connectedness and homology groups of these submanifolds in Riemannian spaces of positive (sectional or qth Ricci) curvature, in particular, spherical and projective spaces. We apply these results to submanifolds with ‘small’ normal curvature, $k_n\le\varepsilon$, and for submanifolds with extrinsic curvature $\le\varepsilon^2$ (resp., non-positive) and small codimension, and include some illustrative examples. The results of the paper generalize theorems about totally geodesic, minimal and k-saddle submanifolds by Frankel; Borisenko, Rabelo and Tenenblat; Kenmotsu and Xia; Mendonça and Zhou.