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Given an Abelian semi-group (A, +), an A-valued curvature measure is a valuation with values in A-valued measures. If A = R, complete classifications of Hausdorff-continuous translation-invariant SO(n)-invariant valuations and curvature measures were obtained by Hadwiger and Schneider, respectively. More recently, characterisation results have been achieved for curvature measures with values in A = Sym^p R^n and A = Sym^2 Λ^q R^n for p, q ≥ 1 with varying assumptions as for their invariance properties.
In the present work, we classify all smooth translation-invariant SO(n)-covariant curvature measures with values in any SO(n)-representation in terms of certain differential forms on the sphere bundle S R^n and describe their behaviour under the globalisation map. The latter result also yields a similar classification of all continuous SO(n)-module-valued SO(n)-covariant valuations. Furthermore, a decomposition of the space of smooth translation-
invariant scalar-valued curvature measures as an SO(n)-module is obtained. As a corollary, we construct explicit bases of continuous translation-invariant scalar-valued valuations and smooth translation-invariant scalar-valued curvature measures.