  Employing a class of generalized connections, we describe certain
differential complices $\left(\tilde \Omega^*_{\mathbb{T}}(M),
\tilde{\mathbb{d}}^{\mathbb{T}}\right)$ constructed from $\wedge^* \mathbb{T}
M$ and study some of their basic properties, where $\mathbb{T} M = T M \oplus
T^*M$ is the generalized tangent bundle on $M$. A number of classical geometric
notions are extended to $\mathbb{T} M$, such as the curvature tensor for a
generalized connection. In particular, we describe an analogue to the
Levi-Civita connection when $\mathbb{T} M$ is endowed with a generalized metric
and a structure of exact Courant algebroid. We further describe in generalized
geometry the analogues to the Chern-Weil homomorphism, a Weitzenb\"ock
identity, the Ricci flow and Ricci soliton, the Hermitian-Einstein equation and
the degree of a holomorphic vector bundle. Furthermore, the Ricci flows are put
into the context of geometric Lax flows, which may be of independent interest.
