# GEOMETRY 2 UNIT 2°

## Learning outcomes of the course unit

The general aim of the course is to develop skills that will allow the students to read and understand mathematical texts, and to reason mathematically with a clear identification of hypothesis and conclusions.

In particular, the course will introduce the theory of curves, some basic notions of holomorphic functions in one complex variables and the basis of differential geometry.

## Prerequisites

Algebra, Analisi 1, Geometria 1.

## Course contents summary

Geometry of curves and surfaces in the tridimensional euclidean space. Basic properties of holomorphic functions of one complex variable.

## Course contents

1. CURVES

1.1 Parametrized curves.

1.2 Regular curves. Arc lenght.

1.3 Local theory of curves. Curvature and torsion.

1.4 Canonical form.

2. SURFACES

2.1 Regular surfaces. Inverse images of regular values. Change of parameters.

2.2 Tangent plane. Firs and second fundamental form of a surface. Normal curvature.

2.3 Orientable surfaces. Gauss map.

2.4 Geometry of the Gauss map. Principal curvatures. Lines of curvature. Mean curvature and Gaussian curvature.

3. INTRINSIC GEOMETRY OF SURFACES

3.1. Isometries. Local isometries.

3.2 Gauss Egregium Theorem. Fundamental equations of a surface.

3.3 Paralle transport. Geodesics.

4. FIRST PROPERTIES OF HOLOMORPHIC FUNCTIONS OF ONE COMPLEX VARIABLES.

4.1 Elemetary functions: polynomial and rational functions, exponential, logarithm function, trigonometric functions. Limits and continuity.

4.2 Complex derivative. Cauchy-Riemann equations.

4.3 Cauchy Theorem. Cauchy Formula. Cauchy inequalities.

## Recommended readings

[1] M. Abate, F. Tovena, Curves and Surfaces, Unitext, 55, Springer, Milano, 2012.

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

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

[4] M. P. Do Carmo, Differential Geometry of Curves and Surfaces, Dover Publications, 2016.

## 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.