About topology of saddle submanifolds

The (k,ε)(k,ε)-saddle (in particular, k  -saddle, i.e. ε=0ε=0) submanifolds are defined in terms of eigenvalues of the second fundamental form. This class extends the class of submanifolds with extrinsic curvature bounded from above, i.e. ⩽ε2⩽ε2 (in particular, non-positive) and small codimension. We study s-connectedness and (co)homology properties of compact submanifolds with ‘small’ normal curvature and saddle submanifolds in Riemannian spaces of positive (sectional or qth Ricci) curvature. The main results are that a submanifold or the intersection of two submanifolds is s-connected under some assumption. By the way, theorems by T. Frankel and some recent results by B. Wilking, F. Fang, S. Mendonça and X. Rong are generalized.