  This paper first introduces the notion of a Rota-Baxter operator (of weight
$1$) on a Lie group so that its differentiation gives a Rota-Baxter operator on
the corresponding Lie algebra. Direct products of Lie groups, including the
decompositions of Iwasawa and Langlands, carry natural Rota-Baxter operators.
Formal inverse of the Rota-Baxter operator on a Lie group is precisely the
crossed homomorphism on the Lie group, whose tangent map is the differential
operator of weight $1$ on a Lie algebra. A factorization theorem of Rota-Baxter
Lie groups is proved, deriving directly on the Lie group level, the well-known
global factorization theorems of Semenov-Tian-Shansky in his study of
integrable systems. As geometrization, the notions of Rota-Baxter Lie
algebroids and Rota-Baxter Lie groupoids are introduced, with the former a
differentiation of the latter. Further, a Rota-Baxter Lie algebroid naturally
gives rise to a post-Lie algebroid, generalizing the well-known fact for
Rota-Baxter Lie algebras and post-Lie algebras. It is shown that the
geometrization of a Rota-Baxter Lie algebra or a Rota-Baxter Lie group can be
realized by its action on a manifold. Examples and applications are provided
for these new notions.
