# GEOMETRY 3

## Learning outcomes of the course unit

The object of the course is to familiarize the students with the basic language of and some fundamental theorems in Differential Geometry, focusing on the coohomological aspects.

## Prerequisites

Analysis 1, 2, Geometry 1, 2, Algebra.

## Course contents summary

Differential Geometry.

## Course contents

1. Manifolds.

1.1 Topological preliminaries.

1.2 Differentiable manifolds, examples.

1.3 Tangent space. Differentiable maps. Differential of a map..

1.4 Vector fields.

1.5 Submanifolds.

2. Tensors and differential forms.

2.1 Tensor algebra.

2.2 Tensor bundles. Differential forms. Exterior derivative.

2.3 Lie derivative.

3. Integration theory on manifolds.

3.1 Oriented manifolds.

3.2 Integrals of differential forms.

3.3 Stokes Theorem.

4. de Rham cohomology and Hodge theory.

4.1 The de Rham complex. Cohomology groups.

4.2 Poincare' Lemma.

4.3 the Hodge star opertaor .

4.4 IHodge Theorem. Poincare' duality.

4.5 Applications of Hodge Theorem.

5. Introduction to Lie groups and Lie algebras. Preliminary notions of Riemannian Geometry.

5.1 Lie groups and LIe algebras: examples.

5.2 The LIe algebra of a LIe group. Exponential mapping.

5.3 Matrix groups.

5.4 Riemannian metrics. Affine connections. Levi-Civita connection. Riemann curvature. Ricci curvature.

5.5 Invariant metrics on LIe groups and curvature properties.

6. Topics on holomrphic functions of one complex variable. olomorfe di una variabile complessa.

6.1. Liouville Theorem.

6.2 Fundamental Theorem of Algebra.

6.3 Laurent series. Residues Theorem.

6.4. Open mapping Theorem.

6.5. meromorphic functions.

6.6. Conformal maps.

7. Topics on algebraic topology.

7.1 Topological coverings.

7.2 Covering transformations.

7.3 Coverings and fundamental groups.

7.4 Universal covering.

References:

[1] W. Boothby, An introduction to differentiable manifolds and Riemannian geometry. Second edition. Pure and Applied Mathematics, 120. Academic Press, Inc., Orlando,

FL, 1986. xvi+430 pp.

[2] F. W. Warner, Foundations of differentiable manifolds and Lie groups. Corrected reprint of the 1971 edition. Graduate Texts in Mathematics, 94. Springer-Verlag, New

York-Berlin, 1983. ix+272

## Recommended readings

[1] W. Boothby, An introduction to differentiable manifolds and Riemannian geometry. Second edition. Pure and Applied Mathematics, 120. Academic Press, Inc., Orlando,

FL, 1986. xvi+430 pp.

[2] F. W. Warner, Foundations of differentiable manifolds and Lie groups. Corrected reprint of the 1971 edition. Graduate Texts in Mathematics, 94. Springer-Verlag, New

York-Berlin, 1983. ix+272

[3] R. V. Churchill, Introduction to Complex Variables and Applications, McGraw- Hill Book Company, Inc., New York, 1948. vi+216 pp.

[4] H. Cartan, Elementary theory of analytic functions of one or several complex variables, Dover Publications, Inc., New York, 1995. 228 pp.

## Teaching methods

Lectures and classroom exercises.

During lectures in traditional mode, the

topics will be formally and rigorously presented. The course will give particular emphasis to application and computation aspects, while not letting out

the theoretical aspect. The classroom exercises are aimed at showing how and where the abstract results can be applied to make the students understand better the relevance of what they are studying.

## Assessment methods and criteria

The exam consists of a written part and an oral part in different dates.