# ELEMENTS OF ADVANCED GEOMETRY

**Academic Year 2023/2024**- Teacher:

**Francesco RUSSO**

## Expected Learning Outcomes

The aim of the course is to allow the students to master the theory and the techniques concerning the differential geometry of curves and surfaces, the geometry of compact connected topological and Riemann surfaces, from a local and global viewpoint.

The students will learn to apply these theories and techniques to the resolution of abstract and concrete problems, which will be assigned through various lists of exercises to be discussed together with the teacher while presenting the solution at the blackboard.

At the end of the course the students will be able to understand the statements and the proof of fundamental theorems of Differential Geometry about gaussian curvature of surfaces and the Teorema Egregium; the local and global theory of holomorphic maps between compact connected Riemann surfaces and the classification of the holorphic structures on a one dimensional torus; local and global properties of surfaces; covariant derivative, affine and riemann connections; geodesics, exponential map and completeness,

## Course Structure

The course consists of theoretical lectures by the teacher and of exercises and worked examples by the teacher and

by the students.

The exercise sessions contemplate a cooperative participation by the students through the exectution of simple calculations

or immediate deductions in order to verify the level of understanding of the theoretical lectures and to test how they are studying

the theoretical arguments via concrete examples, assuring both the assimiliation of the contents of the course and

their ability in solving concrete problems. This would serve also to provide a full preparation for the final oral examination.

Learning assessment may also be carried out on line, should the conditions require it.

PLEASE NOTE: Information for students with disabilities and / or SLD

To guarantee equal opportunities and in compliance with the laws in force, the interested students can ask for a personal interview in order to program any compensatory and / or dispensative measures, according to the didactic objectives and specific needs.

It is also possible to contact the referent teacher CInAP (Centro per Active and Participated Integration - Services for Disabilities and / or DSA) of our Department, prof. Filippo Tired

## Required Prerequisites

Geometria II.

Although not officially required, we highly recommended the parallel course : Complex Analysis and Integral Transforms.

## Attendance of Lessons

## Detailed Course Content

Definition and examples of parametrized curves in the plane and in euclidean spaces in $\mathbb E^n$. Definition of differentiable curve and local theory: arclength, curvature, torsion, Frenét frame and formulas.

Reminder of some facts of general Topology. Definition of topological and differentiable variety. First example. Differentiable surfaces in euclidean space $\mathbb E^3$ and in $\mathbb E^n$. Orientable surfaces. Tangent plane at a point to a differentiable surface. Vector fields tangent to a surface. First and Second Fundamental Theorem of a surface. Isometries between surfaces. Shape Operator and Second Fundamental Form. Curvature of a surface at a point and Gauss Teorema Egregium.

Derivation of a vector field along a curve in space. Introduction to affine and Riemann connections. Parallel transport and covariant derivative.Riemann proof of Gauss Teorema Egregium.

Topological coverings. Connected topological coverings of a simply connected space. Lift of a homotopy between paths. Riemann surfaces and holomorphic maps between Riemann surfaces. Normal forms and applications. Regular value, surjective local diffeomorphisms, ramification and branch point of a holomorphic map between Riemann surfaces.Meromorphic functions on a Riemann surface. Example of $S^2=\mathbb P^1_\mathbb C.$. Field of meromorphic functions on a Riemann surface. Riemann Existence Theorem and applications to the the structure of the field of meromorphic functions on a compact Riemann surface. Isomorphisms between one dimensional complex tori and their structure.Endomorphisms and automorphisms of complex tori. Unirational and rational curves. Luroth Theorem. Elliptic functions as meromorphic functions on the associated torus. Weierstrass p and p' functions. Period map for one dimensional tori. Structure of the field of meromorphic functions of a one dimensional complex torus. Weierstrass canonical form and embedding as complex cubic curve in the projective plane. Riemann Lemma and extension of biholomorphisms between compact Riemann surfaces minus a finite number of points. Euler-Poincaré characteristic of a triangulation of a connected compact topological surface. Radò Theorem. Independence of the Eulero-Poincaré characteristic of a triangulation. Classification of compact, connected topological surfaces. Genus of a compact connected orientable surface. Riemann-Hurwitz and Adjunction Formula.

Geodesics: definition and examples. Exponential map. Complete surfaces. Local form of the Gauss-Bonnet formula for infinitesimal geodetic triangles and global form for compact connected surfaces. Global Theory of surfaces in euclidean space: surfaces of positive, negative and null constant curvature. Hilbert Theorem and impossibility of embedding the Poincaré Half Plane isometrically in euclidean space. Liebmann and Hadamard Theorems.

## Textbook Information

[0] F. Russo, Notes of the Course "Elements of Advanced Geometry", PDF freely available on request, 2022.

[1] W. Boothby, An introduction to differentiable manifolds and Riemannian Geometry, Academic Press, 1986.

[2] S. Kobayashi, Differential Geometry of Curves and Surfaces, Springer, 2019

[3] E. Sernesi, Geometria 2, Bollati Boringhieri, 1994.

[4] K. Tapp, Differential Geometry of Curves and Surfaces, Springer, 2016.

## Learning Assessment

### Learning Assessment Procedures

The exam consist of an oral interview dealing with all the contents of the course.

The rigorous solution of the exercises in the lists will allow the student to apply in explicit examples the powerful techniques learned and will be a basis of discussion during the exam.

The oral exam needs a clear and coincise exposition of the theoretical tools developed during the course in order to verify the process of learning of the student and to prepare him/her to more advanced and specialized courses. Moreover, it is aimed to evaluate the preparation, their expository ability and their personal elaboration of the contents.