= Generalized geometry
Generalized geometry studies structures on the <generalized tangent bundle> $TM\oplus T^*M$, combining <tangent vectors> and <covectors> on a <smooth manifold> $M$. It has a natural split-signature pairing
$$
\langle X+\xi,Y+\eta\rangle=\frac12\bigl(\xi(Y)+\eta(X)\bigr).
$$
No <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 $X+\xi\mapsto X+\xi+\iota_XB$ 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.<Lectures on generalized geometry by Nigel Hitchin>[Sections1.1–1.2 and3.1]{ref}
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