Hopf Galois extensions of Hopf algebroids

We study Hopf Galois extensions of Hopf algebroids as a generalization of the theory for Hopf algebras. More precisely, we introduce (skew-)regular comodules and generalize the structure theorem for relative Hopf modules. Also, we show that if $N\subseteq P$ is a left $\mathcal{L}$-Galois extension and $\mathrm{\Gamma }$ is a 2-cocycle of $\mathcal{L}$, then for the twisted comodule algebra ${}_{\mathrm{\Gamma }}P$, $N\subseteq {}_{\mathrm{\Gamma }}P$ is a left Hopf Galois extension of the twisted Hopf algebroid ${\mathcal{L}}^{\mathrm{\Gamma }}$. We study twisted Drinfeld doubles of Hopf algebroids as examples for the Drinfeld twist theory. Finally, we introduce cleft extension and $\sigma$-twisted crossed products of Hopf algebroids. Moreover, we show the equivalence of cleft extensions, $\sigma$-twisted crossed products, and Hopf Galois extensions with normal basis properties, which generalize the theory of cleft extensions of Hopf algebras.