Transverse Weitzenböck formulas and de Rham cohomology of totally geodesic foliations

Abstract

We prove transverse Weitzenböck identities for the horizontal Laplacians of a totally geodesic foliation. As a consequence, we obtain nullity theorems for the de Rham cohomology assuming only the positivity of curvature quantities transverse to the leaves. Those curvature quantities appear in the adiabatic limit of the canonical variation of the metric.

A horizontal Chern–Gauss–Bonnet formula on totally geodesic foliations

“…Finally, in Sect. 4 we study the small-time asymptotics of Str(e t H,ε ) and conclude the proof of Theorem 1.1.…”

“…4, but we point out that a remarkable feature of that result is that the density ωε H ∧ det T sinh(T ) 1/2 m essentially only depends on horizontal curvature quantities. Therefore, the theorem illustrates further the fact already observed in [4] that topological properties of M might be obtained from horizontal curvature invariants only provided that the bracket-generating condition of the horizontal distribution is satisfied; thus, in essence, the theorem is a sub-Riemannian result. We also note that the condition (1.1) is satisfied in a large class of examples including the H-type foliations introduced in [5], see Example 2.4.…”

“…In this section, we first recall the framework and notations of Baudoin and Grong [4] and the references therein to which we refer for further details. We then prove a necessary and sufficient condition for the form horizontal Laplacian of a totally geodesic foliation to be a symmetric operator.…”

“…The proof of Theorem 1.1 is based on the study of the heat semigroup generated by the hypoelliptic sub-Laplacian on forms recently introduced in [4]. The heat equation approach to Chern-Gauss-Bonnet type formulas (or index formulas) that we are using is of course not new: It was suggested by Atiyah-Bott [1] and McKean-Singer [16] and first carried out by Patodi [18] and Gilkey [12] and is by now classical, see the book [9].…”

Affine Ricci solitons associated to the Bott connection on three-dimensional Lorentzian Lie groups

“…In [1,4,13], the definition of the Bott connection was introduced. In [5], F. Baudoin and E. Grong proved transverse Weitzenb ö ck identities for the horizontal Laplacians of a totally geodesic foliation.…”