Course Outline
0. General theory. (5 lectures). Notes for Lectures 1-5.
L1. Motivation. Definition of G-structure. Examples.
L2. Connections on fiber bundles, vector bundles, principal bundles.
L3. Tautological form. Flat G-structure. Spencer map.
L4. Intrinsic torsion. Prolongation of a G-structure. Spencer cohomology.
L5. General holonomy. Relation to G-structures. Riemannian holonomy. Berger's list.
1. Complex, Kähler, Calabi-Yau geometries. (6 lectures)
L6. GL(n; C)-structures. Complex linear algebra. Newlander-Nirenberg. Dolbeault cohomology.
L7. Holomorphic vector bundles. U(n)-structures. Hermitian linear algebra.
L8. Lefschetz decomposition. Gray-Hervella decomposition. Curvature of Kähler manifolds.
L9. Hodge theory of Hermitian manifolds. Kähler identities and consequences. Hard Lefschetz.
L10. Holomorphic Hermitian vector bundles: Hodge theory, Chern connection. The Bochner method.
L11. Ricci form. SU(n)-structures. Calabi-Yau manifolds.
2. Exceptional Geometries: Oriented Riemannian 4-manifolds; G2 Geometry. (3 lectures)
L12. Oriented 4-Manifolds: Curvature and topology. Hitchin-Thorpe Inequality. K3 surfaces.
L13. Oriented 4-Manifolds: Intro to Twistor theory. Intro to Yang-Mills theory.
L14. G2-structures. Lefschetz decomposition. Fernández-Gray decomposition. Examples.
Logistics
Tue & Thu, 14:00 -- 14:55
March 15 -- May 10
NTU Cosmology Building
(No lecture: Apr 5, May 3-5)
Primary References
[Bes] Arthur L. Besse. Einstein Manifolds. 1987.
[J] Dominic D. Joyce. Riemannian Holonomy Groups and Calibrated Geometry. 2006.
[Sal1] Simon Salamon. Riemannian Geometry and Holonomy Groups. 1989.
Additional References (textbooks)
[K] Shoshichi Kobayashi. Transformation Groups in Differential Geometry. 1972.
[Huy] Daniel Huybrechts. Complex Geometry: An Introduction. 2005.
[M] Andrei Moroianu. Lectures on Kahler Geometry. 2007.
[P] Walter A. Poor. Differential Geometric Structures. 1981.
[Ste] Shlomo Sternberg. Lectures on Differential Geometry. 1964.
Research & Survey Articles
[AHS] M. F. Atiyah, FRS N. J. Hitchin, I. M. Singer. Self-Duality in Four-Dimensional Riemannian Geometry. 1978.
[Bry] Robert Bryant. Metrics with Exceptional Holonomy. 1987.
[ES] J. Eells, S. Salamon. Twistorial Constructions of Harmonic Maps of Surfaces into Four-Manifolds. 1985.
[FFS] M. Falcitelli, A. Farinola, S. Salamon. Almost-Hermitian Geometry. 1994.
[For] Franc Forstneric. The Calabi-Yau property of superminimal surfaces of self-dual Einstein 4-manifolds. 2020.
[G] Victor Guillemin. The Integrability Problem for G-Structures. 1965.
[GH] Alfred Gray, Luis M. Hervella. The Sixteen Classes of Almost Hermitian Manifolds and their Linear Invariants. 1980.
[Hit] Nigel Hitchin. Compact four-dimensional Einstein manifolds. 1974.
[L] Naichung Conan Leung. Geometric Structures on Riemannian Manifolds. 2011.
[Sal2] Simon Salamon. Differential Geometry of Quaternionic Manifolds. 1986.
[SS] I. M. Singer, Shlomo Sternberg. The Infinite Groups of Lie and Cartan, Part I. 1964.