2. Professors
  3. VITAGLIANO Luca
  4. Teaching

FUNDAMENTALS OF HIGHER GEOMETRY

Luca VITAGLIANO FUNDAMENTALS OF HIGHER GEOMETRY

MT22200013
DEPARTMENT OF MATHEMATICS
EQF7
MATHEMATICS
2025/2026



YEAR OF DIDACTIC SYSTEM 2025
AUTUMN SEMESTER
CFUHOURSACTIVITY
648LESSONS
LUCA VITAGLIANO T Curriculum
Objectives
GENERAL AIMS - THE AIM OF THE COURSE IS TO PROVIDE THE FUNDAMENTALS 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 THE FUNDAMENTALS OF THE THEORY OF VECTOR FIELDS AND DIFFERENTIAL FORMS ON SMOOTH MANIFOLDS AND WILL UNDERSTAND THE ROLE OF DIFFERENTIAL GEOMETRY IN CONTEMPORARY MATHEMATICS. MOREOVER, THEY WILL BE ABLE TO INDEPENDENTLY UNDERSTAND THE DEFINITIONS AND FIRST PROPERTIES OF THE GEOMETRIC STRUCTURES THAT CAN DECORATE A SMOOTH MANIFOLD, INCLUDING STRUCTURES WHICH ARE NOT PART OF THE COURSE PROGRAM, FOR INSTANCE RIEMANNIAN, SYMPLECTIC, COMPLEX, CONTACT STRUCTURES, ETC.
APPLICATION OF KNOWLEDGE AND UNDERSTANDING - THE AIM OF THE COURSE IS TO ENABLE THE STUDENT TO APPLY NOTIONS AND TECHNIQUES OF DIFFERENTIAL GEOMETRY IN BOTH GEOMETRIC AND INTERDISCIPLINARY SITUATIONS, WITH A SPECIAL EMPHASIS ON MATHEMATICAL 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. THEY WILL ALSO BE ABLE TO APPLY THE GEOMETRIC METHOD TO THE TREATMENT OF SIMPLE ORDINARY DIFFERENTIAL EQUATIONS E TO MODELING IN CLASSICAL MECHANICS.
AUTONOMY OF JUDGMENT - AT THE END OF THE COURSE THE STUDENT WILL BE ABLE TO DISTINGUISH THE GEOMETRIC STRUCTURES BEHIND SOME SIMPLE DIFFERENTIAL EQUATIONS, BOTH ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS, WITH A SPECIAL EMPHASIS ON EQUATIONS OF CLASSICAL MECHANICS. THEY WILL ALSO BE ABLE TO DISTINGUISH WHICH ASPECTS OF A DYNAMICAL SYSTEM IN CLASSICAL PHYSICS DO DEPEND ON THE CHOICE OF COORDINATES AND WHICH ONES DO NOT, AND, IN THIS SENSE, ARE INTRINSIC.
COMMUNICATION SKILLS - SOLVING THE WEEKLY HOME-WORKS, AND POSSIBLY DISCUSSING THEIR SOLUTIONS WITH THE INSTRUCTOR, THE STUDENT WILL LEARN TO COMMUNICATE IN A CLEAR, CONCISE AND RIGOROUS WAY, BOTH ORALLY AND IN WRITING, THE PROOFS OF SIMPLE (GEOMETRIC) ASSERTIONS THAT THEY HAVE INDEPENDENTLY ELABORATED. MOREOVER, PREPARING THE ORAL TEST, THE STUDENT WILL LEARN HOW TO EXPRESS IN ORGANIC AND COMPLETE WAY, AND USING THE APPROPRIATE “LITERATE REGISTER”, NOT ONLY THE SINGLE DEFINITIONS AND PROPOSITIONS DISCUSSED DURING THE LECTURES, BUT ALSO THE LOGICAL CONNECTIONS BETWEEN DIFFERENT PARTS OF THE PROGRAM AND, MORE GENERALLY, THE STRUCTURE OF THE THEORY.
LEARNING SKILLS - AT THE END OF THE COURSE THE STUDENT WILL BE ABLE TO STUDY ON THEIR OWN ANY TEXTBOOK ON CONTEMPORARY DIFFERENTIAL GEOMETRY, INCLUDING TEXTBOOKS WHICH ARE ADVANCED BOTH IN THE LANGUAGE AND THE CONTENTS, AND THEY WILL BE ABLE TO FILL POSSIBLE GAPS IN THE PROOFS OF THE SIMPLEST ASSERTIONS ALONG THE TEXT THAT THEY ARE STUDYING. THEY WILL ALSO BE ABLE TO CARRY OUT BIBLIOGRAPHIC SEARCHES WITH THE AIM OF FINDING MATERIAL ON COMPLEX AND SPECIALISTIC TOPICS IN DIFFERENTIAL GEOMETRY TO PREPARE A SEMINAR AND/OR AN ESSAY AND/OR A LONGER TEXT (E.G. A GRADUATION THESIS).
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 RING THEORY.
Contents
1.  SMOOTH MANIFOLDS (5 HRS).
CHARTS AND ATLASES. SMOOTH STRUCTURES. SMOOTH MANIFOLDS. TOPOLOGICAL PROPERTIES OF MANIFOLDS. EXAMPLES.

2.  SMOOTH MAPS (7 HRS).
SMOOTH FUNCTIONS ON A MANIFOLD. GLUING LEMMA. SMOOTH MAPS. PULL-BACK. DIFFEOMORPHISMS. SUBMANIFOLDS.

3.  TANGENT SPACES (8 HRS).
TANGENT VECTORS. TANGENT VECTOR THEOREM. TANGENT MAPS. COORDINATE CHANGE. THE TANGENT BUNDLE AND ITS SECTIONS.

4.  IMMERSIONS, SUBMERSIONS, EMBEDDING AND SUBMANIFOLDS (6 HRS).
RANK OF A SMOOTH MAP. IMMERSIONS, SUBMERSIONS AND LOCAL DIFFEOMORPHISMS. RANK THEOREM. IMMERSED SUBMANIFOLDS AND EMBEDDINGS.

5.  VECTOR FIELDS AND FLOWS (10 HRS).
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 (12 HRS).
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. DURING THE COURSE THE LECTURER WILL BE WILLING TO CHECK THE HOMEWORK SOLUTIONS DURING THEIR OFFICE HOURS.
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 LAST AROUND 1 HOUR, AND WILL CONSIST OF THREE TESTS:
1. AN ORAL DISCUSSION ABOUT THE "HOMEWORK" (DURATION AROUND 15 MINUTES), AIMING AT ASSESSING THE AUTONOMY OF JUDGEMENT OF THE STUDENT REGARDING THE COURSE CONTENT. THIS FIRST TEST WILL FOCUS ON 1-2 EXERCISES AMONG THOSE ASSIGNED DURING THE COURSE.
2. SOLVING FEW NEW EXERCISES (DURATION AROUND 10 MINUTES), AIMING AT ASSESSING THE STUDENT ABILITY TO APPLY THE THEORY IN CONCRETE SITUATIONS OF A COMPUTATIONAL NATURE. THIS SECOND TEST WILL FOCUS ON THE RANK OF A SMOOTH MAP, AND CARTAN CALCULUS, INCLUDING THE PULL-BACK OF A DIFFERENTIAL FORM ALONG A SMOOTH MAP.
3. AN ORAL INTERVIEW (DURATION AROUND 35 MINUTES), AIMING AT ASSESSING THE STUDENT'S KNOWLEDGE OF THE THEORETICAL CONTENTS OF THE COURSE. THIS THIRD TEST WILL FOCUS ON THE WHOLE PROGRAM, AND WILL CONSIST OF THREE QUESTIONS. SPECIFICALLY, THE STUDENT WILL TYPICALLY START BY ILLUSTRATING A TOPIC OF THEIR CHOICE AMONG THOSE DISCUSSED DURING THE LECTURES. THE REMAINING TWO QUESTIONS WILL CONCERN THE PARTS OF THE PROGRAM WHICH THE STUDENT HAS NOT YET ILLUSTRATED. IN THIS REPSETC THE PROGRAM MUST BE CONSIDERED AS DIVIDED IN THREE MAIN PARTS: 1) SMOOTH MANIFOLDS AND SMOOTH MAPS, 2) VECTOR FIELDS, FLOWS AND SYMMETRIES, 3) DIFFERENTIAL FORMS.
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
L. VITAGLIANO, A PRIMER ON SMOOTH MANIFOLDS, WORLD SCIENTIFIC.

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
More Information
ALL THE RELEVANT INFORMATION ABOUT THE COURSE CAN BE FOUND AT THE ADDRESS: HTTP://WWW.DIPMAT2.UNISA.IT/PEOPLE/VITAGLIANO/WWW/ISTITUZIONI.HTML
Lessons Timetable

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