Title: VB-Courant algebroids, E-Courant algebroids and generalized geometry,
Abstract: In this paper, we first discuss the relation between VB-Courant algebroids
and E-Courant algebroids and construct some examples of E-Courant algebroids.
Then we introduce the notion of a generalized complex structure on an 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 $\gl(V)\oplus V$ correspond to
complex Lie algebra structures on V.