# GEOMETRY 2 UNIT 2°

## Learning outcomes of the course unit

The course deals with the introduction of the fundamental concepts and the main results concerning the geometry of curves and surfaces in the euclidean space and the first properties of the holomorphic functions of a complex variable.

In particular, the following notion and basic properties of curves and surfaces and holomorphic functions of one complex variable will be introduced:

1) Curvature and torsion of a curve; first and second fundamental form, mean curvature, Gaussian curvature of a surface:

2) Cauchy-Riemann equations, Cauchy's Theorem and Formula, the principle of analytical extension.

## Prerequisites

Algebra, Analisi 1, Geometria 1.

## Course contents summary

Local theory of curves in Euclidean space. Differential geometry of surfaces in the 3-dimensional Euclidean space. Basic properties of holomorphic functions of one complex variable.

## 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.

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.