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In this paper, first we modify the definition of a Hom-Lie algebroidintroduced by Laurent-Gengoux and Teles and give its equivalent dualdescription. Many results that parallel to Lie algebroids are given. Inparticular, we give the notion of a Hom-Poisson manifold and show that there isa Hom-Lie algebroid structure on the pullback of the cotangent bundle of aHom-Poisson manifold. Then we give the notion of a Hom-Lie bialgebroid, whichis a natural generalization of a purely Hom-Lie bialgebra and a Liebialgebroid. We show that the base manifold of a Hom-Lie bialgebroid is aHom-Poisson manifold. Finally, we introduce the notion of a Hom-Courantalgebroid and show that the double of a Hom-Lie bialgebroid is a Hom-Courantalgebroid. The underlying algebraic structure of a Hom-Courant algebroid is aHom-Leibniz algebra, or a Hom-Lie 2-algebra.

Hom-Lie algebroids, Hom-Lie bialgebroids and Hom-Courant algebroids