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). author: Baudoin, Fabrice and Grong, Erlend and Vega-Molino, Sylvie author_list: - affiliation: [] family: Baudoin given: Fabrice - affiliation: [] family: Grong given: Erlend - affiliation: [] family: Vega-Molino given: Gianmarco citations: - author: MF Atiyah doi: 10.2307/1970694 first-page: '374' issue: '86' journal-title: I. Ann. Math. unstructured: 'Atiyah, M.F., Bott, R.: A Lefschetz fixed point formula for elliptic complexes. I. Ann. Math. 2(86), 374–407 (1967)' volume: '2' year: '1967' - author: F Baudoin doi: 10.1016/j.crma.2007.05.005 first-page: '765' issue: '12' journal-title: C. R. Math. Acad. Sci. 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