# Matematica | FUNDAMENTALS OF HIGHER GEOMETRY

## Matematica FUNDAMENTALS OF HIGHER GEOMETRY

 0522200013 DIPARTIMENTO DI MATEMATICA EQF7 MATHEMATICS 2022/2023

 YEAR OF COURSE 1 YEAR OF DIDACTIC SYSTEM 2018 AUTUMN SEMESTER
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/03 6 48 LESSONS COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
 LUCA VITAGLIANO T
ExamDate
ISTITUZIONI DI GEOMETRIA SUPERIORE12/06/2023 - 09:00
ISTITUZIONI DI GEOMETRIA SUPERIORE03/07/2023 - 09:00
Objectives
THE AIM OF THE COURSE IS PROVIDING THE BASIC ELEMENTS OF MODERN DIFFERENTIAL GEOMETRY, WITH A SPECIAL EMPHASIS ON DIFFERENTIAL CALCULUS ON SMOOTH MANIFOLDS.

- KNOWLEDGE AND UNDERSTANDING: AT THE END OF THE COURSE, THE STUDENT WILL KNOW RUDIMENTS OF THE THEORY OF VECTOR FIELDS AND DIFFERENTIAL FORMS ON MANIFOLDS AND WILL UNDERSTAND THE ROLE PLAYED BY DIFFERENTIAL GEOMETRY IN CONTEMPORARY MATHEMATICS. ADDITIONALLY, HE/SHE WILL BE ABLE TO UNDERSTAND, THROUGH AUTONOMOUS STUDY, DEFINITIONS AND FIRST PROPERTIES OF GEOMETRIC STRUCTURE THAT CAN BE ATTACHED TO A SMOOTH MANIFOLDS, INCLUDING THOSE NOT DISCUSSED DURING THE LECTURES, E.G. RIEMANNIAN, SYMPLECTIC, COMPLEX, CONTACT STRUCTURES, ETC.

- APPLYING KNOWLEDGE AND UNDERSTANDING: THE AIM OF THE COURSE IS MAKING THE STUDENT ABLE TO APPLY DIFFERENTIAL GEOMETRIC NOTIONS AND TECHNIQUES BOTH IN A GEOMETRIC OR INTERDISCIPLINARY FRAMEWORK, PARTICULARLY IN ANALYSIS AND MATHEMATICAL PHYSICS. AT THE END OF THE COURSE, THE STUDENT WILL BE ABLE TO APPLY DIFFERENTIAL CALCULUS TO THE STUDY OF SMOOTH MANIFOLD TOPOLOGY. ADDITIONALLY, HE/SHE WILL BE ABLE TO APPLY GEOMETRIC TECHNIQUES TO THE STUDY OF SIMPLE ORDINARY DIFFERENTIAL EQUATIONS AND TO MODELLING IN CLASSICAL MECHANICS.
Prerequisites
THE ONLY PROPEDEUTICAL KNOWLEDGE REQUIRED IS THAT PROVIDED BY UNDERGRADUATE GEOMETRY, ALGEBRA AND ANALYSIS COURSES. IT'S USEFUL, BUT NOT NECESSARY, SOME KNOWLEDGE OF TOPOLOGY AND RING THEORY.
Contents
1.  SMOOTH MANIFOLDS.
CHARTS AND ATLASES. SMOOTH STRUCTURES. SMOOTH MANIFOLDS. TOPOLOGICAL PROPERTIES OF MANIFOLDS. EXAMPLES.

2.  SMOOTH MAPS.
SMOOTH FUNCTIONS ON A MANIFOLD. GLUING LEMMA. SMOOTH MAPS. PULL-BACK. DIFFEOMORPHISMS. SUBMANIFOLDS.

3.  TANGENT SPACES.
TANGENT VECTORS. TANGENT VECTOR THEOREM. TANGENT MAPS. COORDINATE CHANGE. THE TANGENT BUNDLE AND ITS SECTIONS.

4.  IMMERSIONS, SUBMERSIONS, EMBEDDING AND SUBMANIFOLDS.
RANK OF A SMOOTH MAP. IMMERSIONS, SUBMERSIONS AND LOCAL DIFFEOMORPHISMS. RANK THEOREM. IMMERSED SUBMANIFOLDS AND EMBEDDINGS.

5.  VECTOR FIELDS AND FLOWS.
VECTOR FIELDS AND THEIR ALGEBRAIC PROPERTIES. GLUING KLEMMA FOR VECTOR FIELDS. VECTOR FIELDS AND SMOOTH MAPS. INTEGRAL CURVES. EXISTENCE AND UNIQUENESS. FLOWS. LIE DERIVATIVE. SYMMETRIES AND INFINITESIMAL SYMMETRIES.

6.  THE COTANGENT BUNDLE AND DIFFERENTIAL FORMS.
COVECTORS AND THE COTANGENT BUNDLE. DIFFERENTIAL OF A FUNCTION. PULL-BACK OF COVECTORS. 1-FORMS. ALTRNATING FORMS. GRADED ALGEBRAS. DIFFERENTIAL FORMS. CARTAN CALCULUS.
Teaching Methods
TEACHING WILL CONSIST MAINLY OF FRONT LECTURES. HOWEVER, DURING THE LECTURES, THEY WILL BE ASSIGNED EXERCISES AND PROBLEMS TO BE SOLVED BY THE STUDENTS AS "HOMEWORK", WITH THE AIM OF PROMOTING AN "ACTIVE" (HENCE MORE EFFECTIVE) LEARNING PROCESS, AND THE AUTONOMY OF JUDGMENT ON THE SUBJECT OF THE COURSE.
Verification of learning
THE FINAL TEST AIMS AT CHECKING HOW MUCH THE STUDENT LEARNED ABOUT THE THEORY ILLUSTRATED DURING THE COURSE, AND HOW MUCH THE STUDENT UNDERSTANDS THE ROLE OF THIS THEORY IN CONTEMPORARY MATHEMATICS. IT ALSO AIMS AT CHECKING HOW MUCH THE STUDENT IS ABLE TO APPLY THE THEORY TO SOLVING SIMPLE EXERCISES, ALSO IN DIFFERENT CONTEXTS LIKE ANALYSIS AND MATHEMATICAL-PHYSICS. THE EXAMS WILL CONSIST OF THREE TESTS:
1. AN ORAL DISCUSSION ABOUT THE "HOMEWORK",
2. SOLVING FEW NEW EXERCISES,
3. AN ORAL INTERVIEW.
THE THREE PARTS OF THE EXAM WILL TAKE PLACE AT THE SAME TIME. IN THE FINAL ASSESSMENT, EXPRESSED IN THIRTIES, THE ASSESSMENT OF TESTS 1. AND 2. WILL COUNT FOR 20% WHILE THE ORAL INTERVIEW FOR THE REMAINING 80%. THE CUM LAUDE MAY BE GIVEN TO STUDENTS WHO PROVE TO BE ABLETO APPLY THEIR KNOWLEDGE AUTONOMOUSLY EVEN IN CONTEXTS OTHER THAN THOSE PROPOSED IN THE COURSE.
Texts
LECTURE NOTES PROVIDE BY THE INSTRUCTOR

ANOTHER REFERENCE IS
J. M. LEE, INTRODUCTION TO SMOOTH MANIFOLDS (II EDIZIONE), GRADUATE TEXT IN MATHEMATICS, SPRINGER.
IF NECESSARY, THE STUDENT BACKGROUND CAN BE COMPLEMENTED BY TOPOLOGICAL NOTIONS FROM
J. M. LEE, INTRODUCTION TO TOPOLOGICAL MANIFOLDS (II EDIZIONE), GRADUATE TEXT IN MATHEMATICS, SPRINGER.
THE LECTURE NOTES OF THE COURSE (INCLUDING THE HOMOWORKS) BY THE INSTRUCTOR ARE AVAILABLE AT THE ADDRESS: HTTP://WWW.DIPMAT2.UNISA.IT/PEOPLE/VITAGLIANO/WWW/MAIN.PDF