# Differential Operators on Manifolds with G2-Structure

 dc.contributor.author Suan, Caleb dc.date.accessioned 2020-12-16T21:41:32Z dc.date.available 2020-12-16T21:41:32Z dc.date.issued 2020-12-16 dc.date.submitted 2020-12-13 dc.description.abstract In this thesis, we study differential operators on manifolds with torsion-free G2-structure. In particular, we use an identification of the spinor bundle S of such a manifold M with the bundle R ⊕ T*M to reframe statements regarding the Dirac operator in terms of three other first order differential operators: the divergence, the gradient, and the curl operators. We extend these three operators to act on tensors of one degree higher and study the properties of the extended operators. We use the extended operators to describe a Dirac bundle structure on the bundle T*M ⊕ (T*M ⊗ T*M) = T*M ⊗ (R ⊕ T*M) as well as its Dirac operator. We show that this Dirac operator is equivalent to the twisted Dirac operator DT defined using the original identification of S with R ⊕ T*M. As the two Dirac operators are equivalent, we use the T*M ⊕ (T*M ⊗ T*M) = T*M ⊗ (R ⊕ T*M) description of the bundle of spinor-valued 1-forms to examine the properties of the twisted Dirac operator DT. Using the extended divergence, gradient, and curl operators, we study the kernel of the twisted Dirac operator when M is compact and provide a proof that dim (ker DT) = b2 + b3. en dc.identifier.uri http://hdl.handle.net/10012/16565 dc.language.iso en en dc.pending false dc.publisher University of Waterloo en dc.subject differential geometry en dc.subject G2-structures en dc.subject twisted Dirac operator en dc.subject differential operators en dc.title Differential Operators on Manifolds with G2-Structure en dc.type Master Thesis en uws-etd.degree Master of Mathematics en uws-etd.degree.department Pure Mathematics en uws-etd.degree.discipline Pure Mathematics en uws-etd.degree.grantor University of Waterloo en uws.contributor.advisor Karigiannis, Spiro uws.contributor.affiliation1 Faculty of Mathematics en uws.peerReviewStatus Unreviewed en uws.published.city Waterloo en uws.published.country Canada en uws.published.province Ontario en uws.scholarLevel Graduate en uws.typeOfResource Text en

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