Antonio DE NICOLA | DIFFERENTIAL GEOMETRY
Antonio DE NICOLA DIFFERENTIAL GEOMETRY
cod. 0522200008
DIFFERENTIAL GEOMETRY
| 0522200008 | |
| DEPARTMENT OF MATHEMATICS | |
| EQF7 | |
| MATHEMATICS | |
| 2024/2025 |
| YEAR OF COURSE 2 | |
| YEAR OF DIDACTIC SYSTEM 2018 | |
| AUTUMN SEMESTER |
| SSD | CFU | HOURS | ACTIVITY | |
|---|---|---|---|---|
| MAT/03 | 6 | 48 | LESSONS |
| Objectives | |
|---|---|
| THE COURSE WILL BE FOCUSED ON RIEMANNIAN GEOMETRY WHICH IS A BRANCH OF DIFFERENTIAL GEOMETRY STUDYING A MATHEMATICAL OBJECT — CALLED “RIEMANNIAN MANIFOLD” — THAT MODELS THE IDEA OF “CURVED SPACE” OF ARBITRARY DIMENSION. MORE PRECISELY, A RIEMANNIAN MANIFOLD IS A DIFFERENTIABLE MANIFOLD ENDOWED WITH AN ADDITIONAL STRUCTURE, CALLED A RIEMANNIAN METRIC, THAT CONSISTS IN AN INTERNAL PRODUCT ON THE TANGENT VECTOR SPACE AT EACH POINT OF THE MANIFOLD. THIS INTERNAL PRODUCT CHANGES SMOOTHLY WITH THE POINT. THE RIEMANNIAN METRIC ALLOWS TO DEFINE ON THE MANIFOLD MANY OF THE USUAL GEOMETRIC NOTIONS, SUCH AS ANGLES, DISTANCES AND VOLUMES, THE SHORTEST PATH BETWEEN TWO POINTS (CALLED GEODESICS). MOREOVER, CONCEPTS WHICH CHARACTERIZE CURVED SPACES, SUCH AS METRIC TENSOR AND RIEMANNIAN CURVATURE. THE MAIN OBJECTIVES OF THE COURSE “DIFFERENTIAL GEOMETRY” ARE DESCRIBED AS FOLLOWS. KNOWLEDGE AND UNDERSTANDING. AT THE END OF THE COURSE, THE STUDENT WILL KNOW THE RUDIMENTS OF THE THEORY OF VECTOR BUNDLES AND THE GEOMETRY OF RIEMANNIAN AND PSEUDO-RIEMANNIAN MANIFOLDS. HE/SHE WILL UNDERSTAND THE ROLE OF RIEMANNIAN GEOMETRY IN THE LANDSCAPE OF CONTEMPORARY MATHEMATICS. HE/SHE WILL BE ABLE TO GIVE RIGOROUS PROOFS OF RESULTS CONCERNING OR RELATED TO THE TOPICS ADDRESSED (IDEALLY ALSO DIFFERENT FROM THEOREMS PROVED IN THE LECTURES). APPLYING KNOWLEDGE AND UNDERSTANDING. THE PROFICIENT STUDENT SHOULD BE ABLE TO PRODUCE RIGOROUS DEMONSTRATIONS OF RESULTS CONCERNING OR RELATED TO THE TOPICS COVERED (IDEALLY INCLUDING SMALL COROLLARIES OF THEOREMS DEMONSTRATED IN THE LECTURES). THE STUDENT WILL ALSO NEED TO BE ABLE TO APPLY FUNDAMENTAL NOTIONS ABOUT RIEMANNIAN METRICS, GEODESICS, ETC. ALSO TO DIFFERENT CONTEXTS, ESPECIALLY MATHEMATICAL PHYSICS, AND GLOBAL ANALYSIS. UPON COMPLETION OF THE COURSE, THE STUDENT WILL BE ABLE TO STUDY THE METRIC PROPERTIES OF A RIEMANNIAN MANIFOLD, IDENTIFYING ITS GEODESIC EQUATIONS AND CALCULATING CURVATURE TENSORS. |
| Prerequisites | |
|---|---|
| KNOWLEDGE OF GEOMETRY, CALCULUS AND ALGEBRA COURSES OF THE BACHELOR DEGREE IN MATHEMATICS (OR PHYSICS). FAMILIARITY WITH THE NOTIONS OF DIFFERENTIABLE MANIFOLD, TANGENT VECTOR FIELDS, DIFFERENTIAL K-FORMS STUDIED IN AN INTRODUCTORY COURSE OF DIFFERENTIAL GEOMETRY AS THIS COURSE IS MEANT TO BE A NATURAL PROSECUTION. |
| Contents | |
|---|---|
| VECTOR AND TENSOR BUNDLES (8 HOURS): TENSORS, VECTOR BUNDLES. SECTIONS AND LOCAL FRAMES. MORPHISMS OF VECTOR BUNDLES. TENSOR BUNDLES AND TENSOR FIELDS. RIEMANNIAN METRICS AND CONNECTIONS ON VECTOR BUNDLES (16 HOURS): PSEUDO-RIEMANNIAN METRICS. ISOMETRIES. EXAMPLES. CONNECTIONS. COVARIANT DERIVATIVE ALONG A CURVE. PARALLEL TRANSPORT. LEVI-CIVITA CONNECTION. GEODESICS AND THEIR PROPERTIES (16 HOURS). GEODESICS. EXPONENTIAL MAP. LENGTH OF A CURVE. RIEMANNIAN DISTANCE. GEODESICS ARE LOCALLY MINIMISING CURVES. CURVATURE (8 HOURS). RIEMANNIAN, SECTIONAL, RICCI AND SCALAR CURVATURE. EINSTEIN VARIETIES. SPACES WITH CONSTANT CURVATURE. TIME PERMITTING AND DEPENDING ON THE INTEREST OF THE STUDENTS, MORE ADVANCED OR ADDITIONAL TOPICS SUCH AS HOPF-RINOW'S THEOREM OR LORENTZ METRICS MAY BE COVERED. |
| Teaching Methods | |
|---|---|
| TEACHING WILL BE BASED ON TRADITIONAL LECTURES. NEVERTHELESS, EXERCISES WILL BE PROPOSED DURING LECTURES FOR THE STUDENT TO SOLVE IN THE CLASSROOM OR AS A HOMEWORK, WITH THE AIM OF PROMOTING A FORM OF ACTIVE LEARNING (THUS, MORE EFFECTIVE), BESIDES AUTONOMY OF EVALUATION ON THE TOPIC OF THE COURSE |
| Verification of learning | |
|---|---|
| THE EXAM WILL CONSIST OF AN ORAL INTERVIEW AIMED TO VERIFY THE ACHIEVEMENT OF THE ABOVE ILLUSTRATED TEACHING GOALS OF THE COURSE. DURING THE EXAM, THE STUDENT MUST SHOW THAT HE/SHE UNDERSTOOD THE TOPICS TAUGHT DURING THE COURSE AND TO BE ABLE TO EXPLAIN ALL THE LOGICAL STEPS HE USED TO REACH THE CONCLUSION. THE ABILITY TO PRESENT THE ARGUMENTS IN A CORRECT LANGUAGE AND RELATING KNOWN CONCEPTS AND TOPICS. THE CUM LAUDE MARK MAY BE GIVEN TO STUDENTS WHO PROVE TO BE ABLE TO APPLY THEIR KNOWLEDGE AUTONOMOUSLY EVEN IN CONTEXTS OTHER THAN THOSE PROPOSED IN THE COURSE. TO PASS THE EXAM WITH THE MINIMUM MARK, 18/30, THE STUDENT MUST DEMONSTRATE SUFFICIENT KNOWLEDGE OF THE FUNDAMENTAL CONCEPTS (VECTOR BUNDLES, CONNECTION AND PARALLEL TRANSPORT, GEODESICS, CURVATURE) AND TO HAVE LEARNED AT LEAST SOME BASIC PROOFS. |
| Texts | |
|---|---|
| THE REFERENCE TEXTBOOK (IN ITALIAN) IS M. ABATE, F. TOVENA, GEOMETRIA DIFFERENZIALE. UNITEXT 54. LA MATEMATICA PER IL 3+2. SPRINGER, 2011. AN ALTERNATIVE REFERENCE BOOK IN ENGLISH IS J. LEE – RIEMANNIAN MANIFOLDS: AN INTRODUCTION TO CURVATURE, SPRINGER OTHER TEXTBOOKS M. DO CARMO – RIEMANNIAN GEOMETRY, BIRKAUSER P. PETERSEN – RIEMANNIAN GEOMETRY – SPRINGER |
| More Information | |
|---|---|
| SHOULD YOU NEED ANY CLARIFICATION, PLEASE SEND A MESSAGE TO THE EMAIL: ANDENICOLA@UNISA.IT |
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