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University of Waterloo

1. Suan, Caleb. Differential Operators on Manifolds with G2-Structure.

Degree: 2020, University of Waterloo

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.

Subjects/Keywords: differential geometry; G2-structures; twisted Dirac operator; differential operators

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Suan, C. (2020). Differential Operators on Manifolds with G2-Structure. (Thesis). University of Waterloo. Retrieved from http://hdl.handle.net/10012/16565

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Suan, Caleb. “Differential Operators on Manifolds with G2-Structure.” 2020. Thesis, University of Waterloo. Accessed January 21, 2021. http://hdl.handle.net/10012/16565.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Suan, Caleb. “Differential Operators on Manifolds with G2-Structure.” 2020. Web. 21 Jan 2021.

Vancouver:

Suan C. Differential Operators on Manifolds with G2-Structure. [Internet] [Thesis]. University of Waterloo; 2020. [cited 2021 Jan 21]. Available from: http://hdl.handle.net/10012/16565.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Suan C. Differential Operators on Manifolds with G2-Structure. [Thesis]. University of Waterloo; 2020. Available from: http://hdl.handle.net/10012/16565

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

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