MATH 538 Riemannian Geometry II. Lectures TR 1100-1215 SMLC 124.
Instructors: Lectures: TTh 11:00-12:15 SMLC-124, Dimiter Vassilev, Associate Professor;
Office: SMLC, Office 326; Email: firstname.lastname@example.org
Schedule/ Covered Topics:
1. Connections in a vector bundle.
· Connection matrix, curvature.
· Induced connection.
· Covariant exterior derivative (of vector valued forms). Differential Bianchi identity.
· Parallel transport. Holonomy.
· Linear connections – Bianchi and Ricci identities.
2. Riemannian vector bundles.
· Musical isomorphism. Metric connections.
· Exterior derivative and the inner product.
3. Riemannian manifolds.
· Wedge and interior product on forms. Duality. The Hodge-*, d, δ and Δ.
· The deRham complex.
4. Hodge theory.
· Sobolev spaces.
· Symmetric operators, self-adjoint operators, closed operators, compact operators-resolvent, spectrum. Fredholm operators.
· Linear differential operators on vector bundles.
· The Weitzenböck formula for forms.
· Regularity theory for linear partial differential operators.
· Garding’s inequality. The Hodge decomposition.
5. Harmonic vector fields and forms.
· Harmonic vector fields. De Rham’s theorem.
· Harmonic forms. Bochner’s theorem – the Albanese map, harmonic maps, Biebebach’s theorem.
· The index of Δ and d+δ.
6. Vector bundles and characteristic classes.
· Invariant polynomials. Characteristic class of a vector bundle corresponding to an invariant polynomial.
· More properties of the curvature 2-forms. Homogeneous invariant polynomials.
· Pontrjagin classes.
· Chern classes.
7. Intergable distributions. Frboenius’ theorem using forms.
8. Complex geometry.
· Almost complex structures. The holomorphic tangent space, (1,0) and (0,1) vector fields.
· Holomorphic Frobenius theorem. The Newlander-Nirenberg theorem in the analytic case.
· Examples-projective spaces, Hop variety, algebraic varieties.
· Complex and holomorphic vector bundles. Line bundles.
· Almost Hermitian and Hermitian structures.
· Cauchy-Riemann operators. The Hermitian musical isomorphism. The Chern connection.
· Kahler manifolds.