Jacques Van Wyk
An Introduction to Generalised Geometry
Generalised geometry is a field in differential and complex geometry in which one views the direct sum TM ⊕ T*M instead of TM as the bundle associated to a manifold M. Generalised geometry has seen great success in acting as a unifying framework in which structures defined on TM and T*M can be viewed as specific instances of structures defined on TM ⊕ T*M. For example, almost complex structures and pre-symplectic structures can both be viewed as generalised almost complex structures, a certain kind of automorphism of TM ⊕ T*M.
In this talk, I will give an introduction to generalised geometry. I will show TM ⊕ T*M comes with an intrinsic non-degenerate bilinear form. I will introduce the Dorfman bracket on Γ(TM ⊕ T*M), an analogue of the Lie bracket, which together with the aforementioned bilinear form gives TM ⊕ T*M the structure of a Courant algebroid. I will define generalised almost complex structures in this setting, and show how almost complex structures and pre-symplectic structures can be viewed as generalised almost complex structures. I will introduce generalised metrics and generalised connections, and if time permits, I will discuss integrability of generalised almost complex structures in terms of generalised connections, and/or discuss the analogue of the Levi-Civita connection and what complications it comes with.
MC 5417