Generalized geometry is a novel approach to geometric structures pioneered by Hitchin in 2003. Developed further by Gualtieri, Cavalcanti and others, it soon became an active topic catching the interest and bringing together the expertise of geometers and theoretical physicists. For example, generalized complex geometry provides a unifying framework for complex and symplectic structures. In it, we can smoothly pass from a symplectic to a complex structure and have generalized complex manifolds that admit neither complex nor symplectic structures. The theory is supported by Clifford algebras and the theory of spinors, and built on previous developments on Courant algebroids and Dirac structures, which arose as the mathematical formalism for mechanical systems in the presence of both symmetries and constraints.
In this introductory course we will isolate the underlying linear algebra from the actual geometry and draw parallelism between the classical and the generalized setting. In this sense, we will start with some linear algebra that is not usually taught at the undergraduate level, continue with the linear algebra of generalized geometry, recall the basic facts about differential geometry of manifolds (most importantly, integrability), and finally combine all this knowledge to develop generalized geometry.
We will cover the following topics:
I. Linear algebra
• Linear versions of symplectic, complex and Poisson structures.
• The exterior algebra as a language for linear algebraic structures.
• Spaces of structure and torsors.
II. Generalized linear algebra
• The vector space V⊕V*: the canonical pairing and its orthogonal transformations.
• Isotropic subspaces and annihilators of forms.
• Linear Dirac and generalized complex structures. The type.
• The role of the Clifford algebra and spinors.
III. Geometry
• Manifolds, bundles and Cartan calculus.
• Almost complex structures and non-degenerate 2-forms.
• Integrability: complex and symplectic structures.
• Poisson structures.
IV. Generalized geometry
• The Dorfman bracket and the Courant algebroid TM⊕T*M . The group of generalized diffeomorphisms.
• Dirac structures: geometric interpretation as foliations, type-change structures, integrability in terms of spinors.
• Generalized complex structures: type change, is there anything between complex and symplectic?, is there anything beyond complex and symplectic?
• Other topics, depending on the interest and time: twisted versions, other generalized tangent bundles, generalized riemannian geometry, deformation theory of complex and generalized complex structures, applications to physics, possible research directions.