= Generalized tangent bundle
{title2=$\mathbb TM=TM\oplus T^*M$}
The <generalized tangent bundle> of a <smooth manifold> $M$ is the <direct sum of vector bundles> $\mathbb TM=TM\oplus T^*M$. A fibre contains a <tangent vector> and a <covector> together. In real <dimension> $n$, it has rank $2n$ and a canonical symmetric <bilinear form>
$$
\langle X+\xi,Y+\eta\rangle=\tfrac12\bigl(\xi(Y)+\eta(X)\bigr)
$$
of signature $(n,n)$. No <Riemannian metric> is needed.<Lectures on generalized geometry by Nigel Hitchin>[sections 1.1–1.2]{ref}
Both $TM$ and $T^*M$ are isotropic: the pairing of any two elements of either summand is zero. A <two-form> $B$ gives the transformation $X+\xi\mapsto X+\xi+\iota_XB$. 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.
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