# 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

Anallysis 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.

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

## Teaching methods

Theoretical lectures and sessions of oral and written exercises.

## Assessment methods and criteria

Homeworks and oral exam.