Positive scalar curvature on foliations: The noncompact case

Abstract

Let (M,gTM) be a noncompact (not necessarily complete) enlargeable Riemannian manifold in the sense of Gromov-Lawson and F an integrable subbundle of TM. Let kF be the leafwise scalar curvature associated to gF=gTM|F. We show that if either TM or F is spin, then inf(kF)≤0. This generalizes the famous result of Gromov-Lawson on enlargeable manifolds to the case of foliations. It also extends an ansatz of Gromov on hyper-Euclidean spaces to general enlargeable Riemannian manifolds, as well as recent results on compact enlargeable foliated manifolds due to Benameur-Heitsch et al. to the noncompact situation.