In this paper, we first discuss the relation between $\text{VB}$-Courant algebroids and $\text{E}$-Courant algebroids, and we construct some examples of $\text{E}$-Courant algebroids. Then we introduce the notion of a generalized complex structure on an $\text{E}$-Courant algebroid, unifying the usual generalized complex structures on even-dimensional manifolds and generalized contact structures on odd-dimensional manifolds. Moreover, we study generalized complex structures on an omni-Lie algebroid in detail. In particular, we show that generalized complex structures on an omni-Lie algebra $\text{gl}\left( V \right)\oplus V$ correspond to complex Lie algebra structures on $V$.

To a Lie groupoid over a compact base $M$, the associated group of bisection is an (infinite-dimensional) Lie group. Moreover, under certain circumstances one can reconstruct the Lie groupoid from its Lie group of bisections. In the present article we consider functorial aspects of these construction principles. The first observation is that this procedure is functorial (for morphisms fixing $M$). Moreover, it gives rise to an adjunction between the category of Lie groupoids over $M$ and the category of Lie groups acting on $M$. In the last section we then show how to promote this adjunction to almost an equivalence of categories.

A surface $\sum$ endowed with a Poisson tensor $\pi$ is known to admit a canonical integration, $G\left( \pi \right)$, which is a 4-dimensional manifold with a (symplectic) Lie groupoid structure. In this short note we show that if $\text{ }\!\!\pi\!\!\text{ }$ is not an area form on the 2-sphere, then $G\left( \pi \right)$ is diffeomorphic to the cotangent bundle $T*\sum$. This extends results by the author and by Bonechi, Ciccoli, Staffolani, and Tarlini.

We propose a new, constructive theory of moving frames for Lie pseudo-group actions on submanifolds. The moving frame provides an effective means for determining complete systems of differential invariants and invariant differential forms, classifying their syzygies and recurrence relations, and solving equivalence and symmetry problems arising in a broad range of applications.