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: vassilev@unm.edu

Office Hours:  Feel free to stop-by anytime you have a question.

Catalog Description: Continuation of MATH 537 Riemannian Geometry I with emphasis on adding more structures. Riemannian submersions, Bochner theorems with relation to topology of manifolds, Riemannian Foliations, Complex and Kaehler geometry, Sasakian and contact geometry. Prerequisite: 537.

ATTENDANCE: Attendance at UNM is mandatory, see policy.

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.