Not an answer, only an attempt to clarify.
I'm guessing this is what is meant by "generalized geometry":
"Generalized geometry is based on two premises – the first is to replace the tangent
bundle $T$ of a manifold $M$ by $T \oplus T^*$, and the second to replace the Lie bracket on
sections of $T$ by the Courant bracket. The idea then is to use one’s experience of
differential geometry and by analogy to define and develop the generalized version.
Depending on the object, this may or may not be a fertile process, but the intriguing
fact is that, by drawing on the intuition of a mathematician, one may often obtain
this way a topic which is also of interest to the theoretical physicist."
The quote is from:
Hitchin, Nigel. "Lectures on generalized geometry."
arXiv:1008.0973 (2010).
52 pages; based on JCAS Lecture Series in Hong Kong.