Homology vanishing theorems for submanifolds

We relate intrinsic and extrinsic curvature invariants to the homology groups of submanifolds in space forms of nonnegative curvature. More precisely, we provide bounds for the squared length of the second fundamental form, or the Ricci curvature in terms of the mean curvature, which force homology to vanish in a range of intermediate dimensions. Moreover, we give examples which show that these conditions are sharp.