Generalized tangent bundle ($\mathbb{T}M=TM\oplus T^{\ast }M$)

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.