 Generalized geometry

Generalized geometry studies structures on the generalized tangent bundle $TM\oplus T^{\ast }M$, combining tangent vectors and covectors on a smooth manifold $M$. It has a natural split-signature pairingNo Riemannian metric is needed to define this pairing. The factor one-half is a convention. A Courant bracket combines the Lie bracket of vector fields with derivatives of one-forms, adding differential information to the bundle.

For a two-form $B$, the transformation $X+\xi \mapsto X+\xi +{\iota }_{X}B$ preserves the pairing: the two new cross terms cancel because $B$ is alternating. When $dB=0$, it also preserves the standard Courant bracket. These transformations enlarge the ordinary diffeomorphism symmetries. The framework places complex structures and symplectic forms within the common language of generalized complex geometry.^[ref]