MAT 682 
Advanced Topics in
Differential Geometry 
Fall 2021 

TΘ 11:30-12:50
ESS 181

Prof. Claude LeBrun.
Office: Math Tower 3-108.
Office hours: TΘ 2:30--4:00 pm.

Self-Dual 4-Manifolds

This one-semester course will offer a self-contained introduction to self-dual 4-manifolds and their twistor spaces. Here self-duality is a conformally-invariant property on a Riemannian 4-manifold, roughly meaning that half of its conformal curvature vanishes. This special class of Riemannian 4-manifolds is intimately related to a special class of complex 3-folds by the Penrose twistor correspondence, where the conformal structure of the 4-manifold determines the complex structure of its twistor space, and vice versa. This abstract theory will be carefully illustrated through the exploration of numerous concrete examples.

The lectures will presuppose a basic knowledge of Riemannian geometry, roughly at the level of MAT 568. Some familiarity with complex manifolds, say at the level of MAT 545, could also be extremely helpful.

Grades will be based on attendance and class participation.

The Professor may be reach via e-mail by .
This is the best way to request a Zoom meeting outside of normal office hours.

Illustration: Ruling lines on a quadric surface. The distinction between self-dual and anti-self-dual 2-forms is implicit in this picture.

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