ON SUBMANIFOLDS WITH TAMED SECOND FUNDAMENTAL FORM

Based on the ideas of Bessa, Jorge and Montenegro (Comm. Anal. Geom., vol. 15, no. 4, 2007, pp. 725-732) we show that a complete submanifold M with tamed second fundamental form in a complete Riemannian manifold N with sectional curvature K-N <= kappa <= 0 is proper (compact if N is compact). In addition, if N is Hadamard, then M has finite topology. We also show that the fundamental tone is an obstruction for a Riemannian manifold to be realised as submanifold with tan-led second fundamental form of a Hadamard manifold with sectional curvature bounded below.