Document Type

Thesis

Degree Name

Master of Science (MSc)

Department

Mathematics

Program Name/Specialization

Mathematics for Science and Finance

Faculty/School

Faculty of Science

First Advisor

Shengda Hu

Advisor Role

Research Advisor

Second Advisor

Spiro Karigiannis

Advisor Role

Research Advisor

Abstract

The Atiyah-Hitchin-Singer theorem states that the twistor almost complex structure on a certain S2 bundle over an oriented Riemannian 4-manifold (M, g) is integrable if and only if the Weyl curvature tensor of g is self-dual. These ideas were developed by Roger Penrose connecting 4-dimensional Riemannian geometry with complex geometry. We present a new approach to the Atiyah-Hitchin-Singer theorem using horizontal lifts and their respective flows, cross products and the quaternions to show that the Nijenhuis tensor vanishes if and only if the Weyl curvature tensor of g is anti-self-dual. An eight dimensional generalization is presented when the manifold is R8.

Convocation Year

2024

Convocation Season

Fall

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