Transverse Weitzenböck formulas and curvature dimension inequalities on Riemannian foliations with totally geodesic leaves

Abstract

We prove a family of Weitzenböck formulas on a Riemannian foliation with totally geodesic leaves. These Weitzenböck formulas are naturally parametrized by the canonical variation of the metric. As a consequence, under natural geometric conditions, the horizontal Laplacian satisfies a generalized curvature dimension inequality. Among other things, this curvature dimension inequality implies Li-Yau estimates for positive solutions of the horizontal heat equation, sharp eigenvalue estimates and a sub-Riemannian Bonnet-Myers compactness theorem whose assumptions only rely on the intrinsic geometry of the horizontal distribution.

Transverse Weitzenböck formulas and de Rham cohomology of totally geodesic foliations

“…Proof. As it has been shown in [6] (see also Theorem 4.7 in [3]), the formula (2) implies that for any smooth form α,…”

“…A computation similar to the proof of Proposition 3.6 in [6] shows then that if α is a closed one-form one has then…”

“…In Section 2, we focus first on transverse Weitzenböck formulas on one-forms. We prove that the horizontal Laplacian on one-forms that was first introduced in [6] can actually be written as ∆ H = −δ H d − dδ H where δ H is a horizontal divergence computed with respect to some connnection. Taking then advantage of the Weitzenböck formula proved in [6], this allows us to apply Bochner's technique and obtain a sufficient criterion for the nullity of the first de Rham cohomology group.…”

“…The adjoint connection of the Bott connection is not metric. For this reason, we shall rather make use of the following family of connections first introduced in [4,6]:…”

“…The proposition will be proved in greater generality in Proposition 3.3. We give here a sketch of direct proof that allows to identify explicitly the tensors on 1-forms and comparison to [6]. Let x ∈ M. From [6], around x, there exist a local orthonormal horizontal frame {X 1 , · · · , X n } and a local orthonormal vertical frame {Z 1 , · · · , Z m } such that the following structure relations hold…”