ELEMENTS OF ADVANCED GEOMETRY

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, 

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.

If the course is delivered in blended or remote mode, appropriate adjustments may be made to the above, in order to ensure consistency with the syllabus.

Geometria II. 

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

Highly recommended.

Definition and examples of parametrized curves in the plane and in euclidean spaces in  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   E^3 and in 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. Riemann proof of Gauss Teorema Egregium.

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.

If time allows, some of these arguments will be presented:

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=P^1_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.  

[0] F. Russo,  Notes of the Course "Elements of Advanced Geometry", PDF freely available on request, 2024.

[00] L. Tu, An Introduction to Manifolds, Second Edition, Springer Verlag 2010.
[1] W. Boothby, An introduction to differentiable manifolds and Riemannian Geometry, Academic Press, 1986.
[2] E. Sernesi, Geometria 2, Bollati Boringhieri, 1994.
[3] K. Tapp, Differential Geometry of Curves and Surfaces, Springer, 2016.

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

The exam consists of an oral interview dealing with all the contents of the course. The rigorous solution of the exercises in the various lists will allow the student to apply in explicit examples the powerful techniques learned. Moreover, these examples will serve as a basis for the 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, the expository ability and the personal elaboration of the contents.

Learning assessment may also be carried out on-line, should the conditions require it. To ensure equal opportunities and in compliance with current laws, interested students may request a personal interview in order to plan any compensatory and/or dispensatory measures based on educational objectives and specific needs. Students can also contact the CInAP (Centro per l'integrazione Attiva e Partecipata – Servizi per le Disabilità e/o i DSA) referring teacher within their department (https://www.cinap.unict.it/content/referenti).

The questions will concentrate on ALL the contents of the course  and on the required exercises/examples from  the assigned exercise lists.