Differential Geometry

Al termine del corso   gli studenti saranno in grado di comprendere  enunciati e dimostrazioni di teoremi fondamentali della Geometria Differenziale e della Geometria Riemanniana concernenti la curvatura gaussiana di una superficie e il Teorema Egregium di Gauss; la teoria locale e globale delle geodetiche su una varietà riemanniana; derivazione covariante, connessioni affini e riemanniane; geodetiche, mappa esponenziale e varietà riemanniane complete.

The aim of the course is to allow the students to master the theory and the techniques concerning the differential geometry abstract differentiable manifolds, from a local and global point of view, with emphasis on the case of riemannian manifolds. In particular, the study of vector and tensor fields and of differential forms on differentiable manifolds are introduced and studied, as well the theory of covariant derivative, affine and Riemann connections and the curvature tensor of a Riemannian connection.

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 and fo Riemannian Geometry, including gaussian curvature of surfaces and the Teorema Egregium; the local and global theory of geodesics on Riemannian manifolds; exponential maps and complete riemannian manifolds.

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

Highly recommended but not mandatory: Algebraic Topology.

[0] F. Russo,  Note del Corso  "Geometria Differenziale", PDF freely available on request, 2022.

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

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

[4] L. Tu, An Introduction to Manifolds, Second Edition, Springer, 2010.

[5] L. Tu, Differential Geometry, Springer 2017.

[6] J. M. Lee, Introduction to Riemannian Manifolds, Second Edition, Springer, 2018.

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.