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University of Oxford

1. Rubio, Roberto. Generalized geometry of type Bn.

Degree: PhD, 2014, University of Oxford

Generalized geometry of type Bn is the study of geometric structures in T+T<sup>*</sup>+1, the sum of the tangent and cotangent bundles of a manifold and a trivial rank 1 bundle. The symmetries of this theory include, apart from B-fields, the novel A-fields. The relation between Bn-geometry and usual generalized geometry is stated via generalized reduction. We show that it is possible to twist T+T<sup>*</sup>+1 by choosing a closed 2-form F and a 3-form H such that dH+F2=0. This motivates the definition of an odd exact Courant algebroid. When twisting, the differential on forms gets twisted by d+Fτ+H. We compute the cohomology of this differential, give some examples, and state its relation with T-duality when F is integral. We define Bn-generalized complex structures (Bn-gcs), which exist both in even and odd dimensional manifolds. We show that complex, symplectic, cosymplectic and normal almost contact structures are examples of Bn-gcs. A Bn-gcs is equivalent to a decomposition (T+T<sup>*</sup>+1)<sub>ℂ</sub>= L + L + U. We show that there is a differential operator on the exterior bundle of L+U, which turns L+U into a Lie algebroid by considering the derived bracket. We state and prove the Maurer-Cartan equation for a Bn-gcs. We then work on surfaces. By the irreducibility of the spinor representations for signature (n+1,n), there is no distinction between even and odd Bn-gcs, so the type change phenomenon already occurs on surfaces. We deal with normal forms and L+U-cohomology. We finish by defining G22-structures on 3-manifolds, a structure with no analogue in usual generalized geometry. We prove an analogue of the Moser argument and describe the cone of G22-structures in cohomology.

Subjects/Keywords: 516; Mathematics; 3-manifold; almost contact geometry; complex geometry; deformation theory; G2(2)-structure; generalized complex geometry; twisted cohomology; generalized geometry; Lie algebroid

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APA (6th Edition):

Rubio, R. (2014). Generalized geometry of type Bn. (Doctoral Dissertation). University of Oxford. Retrieved from ;

Chicago Manual of Style (16th Edition):

Rubio, Roberto. “Generalized geometry of type Bn.” 2014. Doctoral Dissertation, University of Oxford. Accessed September 19, 2020. ;

MLA Handbook (7th Edition):

Rubio, Roberto. “Generalized geometry of type Bn.” 2014. Web. 19 Sep 2020.


Rubio R. Generalized geometry of type Bn. [Internet] [Doctoral dissertation]. University of Oxford; 2014. [cited 2020 Sep 19]. Available from: ;

Council of Science Editors:

Rubio R. Generalized geometry of type Bn. [Doctoral Dissertation]. University of Oxford; 2014. Available from: ;