Abstract
Under suitable conditions, we show that the Euler characteristic of a foliated Riemannian manifold can be computed only from curvature invariants which are transverse to the leaves. Our proof uses the hypoelliptic sub-Laplacian on forms recently introduced by two of the authors in Baudoin and Grong (Ann Glob Anal Geom 56(2):403–428, 2019).
Highlights
The goal of the paper is to prove the following result: Theorem 1.1 Let M be a smooth, connected, oriented and n + m dimensional compact manifold
We assume that M is equipped with a Riemannian foliation F with bundle-like metric g and totally geodesic m-dimensional leaves
We assume that the horizontal distribution H = F⊥ is bracket-generating and that there exists ε > 0 such that
Summary
The goal of the paper is to prove the following result: Theorem 1.1 Let M be a smooth, connected, oriented and n + m dimensional compact manifold. We assume that M is equipped with a Riemannian foliation F with bundle-like metric g and totally geodesic m-dimensional leaves. We assume that the horizontal distribution H = F⊥ is bracket-generating and that there exists ε > 0 such that (∇v J )w = − 2ε [Jv, Jw]. For any v, w ∈ Tx M, x ∈ M, where ∇ is the Bott connection of the foliation and J is the tensor defined in (2.2). Denoting χ(M) the Euler characteristic of M:.
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