ON SUBMANIFOLDS WITH TAMED SECOND FUNDAMENTAL FORM

Abstract

Abstract 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 ≤ κ ≤ 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 tamed second fundamental form of a Hadamard manifold with sectional curvature bounded below.