It is of interest to extend classical geometric notions to generalized geometry.
Various approaches have been proposed in the recent literature. Employing a
class of generalized connections, we describe certain differential complices
constructed
from
and study some of their basic properties, where
is the generalized
tangent bundle on
.
To illustrate how various constructions fit together from this point of view, we
describe within the proposed framework the analogues to the Levi-Civita connection
when
is endowed with a generalized metric and a structure of exact Courant algebroid, the
Chern–Weil homomorphism, a Weitzenböck identity, the Ricci flow as a Lax flow
and Ricci soliton, the Hermitian–Einstein equation and the degree of a holomorphic
vector bundle.
