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]

The untwisted skew-symmetric Courant bracket on sections of the generalized tangent bundle $TM\oplus T^{\ast }M$ isHere $[X,Y]$ is the Lie bracket of vector fields, $\mathcal{L}$ is the Lie derivative, and $d$ is the exterior derivative. It extends the Lie bracket of vector fields, but its Jacobi identity has an exact-form defect; it is not simply another Lie algebra bracket.^[ref]

For pure vector fields, $[X,Y{]}_C=[X,Y]$, and two pure one-forms have zero bracket. On the real line, $[{\partial }_x,f(x)dx{]}_C=\frac{1}{2}{f}^{\prime }(x)dx$. This mixed example shows how the bracket couples tangent and cotangent information through differentiation. Omitting the final exact term gives the nonskew Dorfman operation, a related convention with different identities.

The generalized tangent bundle of a smooth manifold $M$ is the direct sum of vector bundles $\mathbb{T}M=TM\oplus T^{\ast }M$. A fibre contains a tangent vector and a covector together. In real dimension $n$, it has rank $2n$ and a canonical symmetric bilinear formof signature $(n,n)$. No Riemannian metric is needed.^[ref]

Both $TM$ and $T^{\ast }M$ are isotropic: the pairing of any two elements of either summand is zero. A two-form $B$ gives the transformation $X+\xi \mapsto X+\xi +{\iota }_{X}B$. The new pairing terms sum to $B(X,Y)+B(Y,X)=0$, so the transformation preserves the pairing. These extra symmetries explain the utility of the bundle in generalized geometry, where complex structures and symplectic forms can be treated in one framework.