Vanishing theorems on foliations

Weiping Zhang

Published 2015-08-19Version 1

We prove the following generalization of the Lichnerowicz-Hitchin vanishing theorem to the case of foliations: let $M$ be a closed spin manfold, let $F$ be an integrable subbundle of the tangent bundle $TM$ such that $F$ carries a metric of positive leafwise scalar curvature, then the canonical $KO$-characteristic number $\hat{\mathcal A}(M)$ vanishes. Our proof applies to give a geometric proof of the Connes vanishing theorem, which states that in the case of $F$ being spin instead of $TM$ being spin, one has $\hat{A}(M)=0$.