2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/119250We study different notions of Riemannian curvatures: The $p$-curvatures which interpolate between the scalar curvature and the sectional curvature, the Gauss-Bonnet-Weyl curvatures form another interpolation from the scalar curvature to the Gauss-Bonnet integrand. We bring out the $(p,q)$-curvatures, which incorporate all the previous curvatures. We then examine the curvature term which appears in the classical Weitzenböck formula. We also study the positivity properties of the $p$-curvatures, the second Gauss-Bonnet-Weyl curvature, the Einstein curvature and the isotropic curvature.In french, extracted from habilitation thesis at Montpellier II University (France), 50 pagesDifferential GeometryMathematical Physics53C21; 53C25; 53B20, 58A14; 58D17; 58E11text