Positive scalar curvature on foliations: the noncompact case
Let $(M,g^{TM})$ be a noncompact enlargeable Riemannian manifold in the sense of Gromov-Lawson and $F$ an integrable subbundle of $TM$. Let $k^{F}$ be the leafwise scalar curvature associated to $g^F=g^{TM}|_F$. We show that if either $TM$ or $F$ is spin, then ${\rm inf}(k^F)\leq 0$. This generalizes earlier claims for hyper-Euclidean spaces made by Gromov.