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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.

Marco Gualtieri works in differential geometry and mathematical physics, with a focus on generalized geometry. His own research page identifies that focus rather than presenting him only as an instructor.^[ref] The geometric viewpoint concerns structures carried by a smooth manifold, where tangent directions and covectors can be studied together. It connects the study of complex geometry and symplectic geometry with mathematical descriptions of physical fields.

Homepage: www.math.toronto.edu/mgualt/

Instructor for the 2012 Topology I: smooth manifolds (MAT1300HF) course; the course information explicitly identifies his role as instructor.