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.