# 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.

The main goals of the course are the study of the elementary properties of the holomorphic functions of a complex variable and of the geometrical properties of the differentiable varieties, with particular reference to the cohomological aspects of them. At the end of the course, students will be familiar with the holomorphic function theory of a complex variable, with the Cauchy formula, with Laurent's deille series theory and with residual calculus, with the differential calculation on real manifolds, with the Hodge theory on compact Riemannian manifolds and with the first elements of Riemannian Geometry. They will also be able to tackle the resolution of problems of a theoretical and practical nature in the field of Differential Geometry and Holomorphic Functions Theory of a Complex Variable.

## 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.

6. Holomorphic Functions of one Complex Variable.

6.1. Functions with complex values. Continuity.

6.2. C-ddifferentiable functions. Holomorphic functions. Conditions of Cauchy-Riemann. The Cauchy-Riemann operator.

6.3. Elementary functions. Polynomials, Rational Functions. Exponential, Logarithm, Power.

7. Complex integration.

7.1. Integral of a complex value function along a path.

7.2. Domains on a regular border.

7.3. Cauchy's theorem.

7.4. The Cauchy formula.

7.5. Cauchy's inequalities.

7.6. Analytical functions. The principle of analytic prolongation. The Liouville Theorem. The Fundamental Theorem of algebra.

8. Laurent series.

8.1. Singularity of a holomorphic function.

8.2. Taylor and Laurent series development.

8.3. Residue Theorem.

8.4. Applications of the theory of residues to the calculation of integrals.

## 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] M. P. Do Carmo, Riemannian Geometry, Birkhäuser, 1992.

[4] Theodore W. Gamelin, Complex Analysis, Springer, 2003.

## Teaching methods

Lectures and exercises

The lessons will be organized as follows:

in person with the possibility of using the lessons also remotely in synchronous mode (via Teams). During the lectures, in traditional mode, the

topics will be presented in a formal and rigorous way. There will be exercises using the elly platform. The course will give particular emphasis to the application and calculation aspects, while not neglecting

the theoretical aspect. For this purpose, the exercises carried out in the classroom in which the student learns to apply the

theory seen in class when solving concrete problems.

## Assessment methods and criteria

The final exam consists of a written test and an oral exam. The evaluation of the written test is as follows: students who score between 24 and 30, achieve A.

Students who score between 18 and 23, achieve B. Students who score below 18, achieve C. The written test is considered passed when at least B.

The oral exam consists in the proofs of significant theorems and / or in the exposition of topics, definitions, treated in the lectures.

If due to the persistence of the health emergency it was

necessary to integrate with the remote modality the carrying out of the exams will proceed as follows:

remote oral questions, remote written tests.

## Other informations

Exercises and problems to be performed outside of class hours will be assigned by the teacher.

Notes from the teacher will be distributed. The course notes in PDF format and all the material used during the lessons and exercises are made available to the students on the Elly educational platform.