Matematica | DIFFERENTIAL GEOMETRY

cod. 0522200008

DIFFERENTIAL GEOMETRY

	0522200008
	DIPARTIMENTO DI MATEMATICA
	EQF7
	MATHEMATICS
	2022/2023

CURRICULUM GENERAL MATHEMATICS AND DIDACTICS OF MATHEMATICS

	YEAR OF COURSE 2
	YEAR OF DIDACTIC SYSTEM 2018
	AUTUMN SEMESTER

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	MAT/03	6	48	LESSONS	SUPPLEMENTARY COMPULSORY SUBJECTS

EXAMS

	Exam	Date	Session
	GEOMETRIA DIFFERENZIALE	12/06/2023 - 09:00	SESSIONE ORDINARIA
	GEOMETRIA DIFFERENZIALE	03/07/2023 - 09:00	SESSIONE ORDINARIA

	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 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 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 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.

	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
	RIEMANN METRICS. ISOMETRIES. EXAMPLES. CONNECTIONS. COVARIANT DERIVATIVE ALONG A CURVE. PARALLEL TRANSPORT. LEVI-CIVITA CONNECTION. GEODESICS. EXPONENTIAL MAP. LENGTH OF A CURVE. RIEMANNIAN DISTANCE. GEODESICS ARE LOCALLY DISTANCE-MINIMIZING CURVES. RIEMANNIAN, RICCI, AND SCALAR CURVATURE. EINSTEIN MANIFOLDS. CONSTANT CURVATURE SPACES. TIME PERMITTING, AND KEEPING INTO ACCOUNT THE INTEREST OF THE STUDENTS, MORE ADVANCED OR ADDITIONAL TOPICS WILL BE TREATED, SUCH AS HOPF-RINOW THEOREM OR LORENTZ METRICS.

Contents

RIEMANN METRICS. ISOMETRIES. EXAMPLES. CONNECTIONS. COVARIANT DERIVATIVE ALONG A CURVE. PARALLEL TRANSPORT. LEVI-CIVITA CONNECTION. GEODESICS. EXPONENTIAL MAP. LENGTH OF A CURVE. RIEMANNIAN DISTANCE. GEODESICS ARE LOCALLY DISTANCE-MINIMIZING CURVES. RIEMANNIAN, RICCI, AND SCALAR CURVATURE. EINSTEIN MANIFOLDS. CONSTANT CURVATURE SPACES. TIME PERMITTING, AND KEEPING INTO ACCOUNT THE INTEREST OF THE STUDENTS, MORE ADVANCED OR ADDITIONAL TOPICS WILL BE TREATED, SUCH AS HOPF-RINOW THEOREM OR LORENTZ METRICS.

	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 LEARNING OF THE THEORY ILLUSTRATED DURING THE COURSE, THE UNDERSTANDING OF ITS ROLE IN THE CONTEMPORARY MATHEMATICS LANDSCAPE, AND THE ABILITY OF THE STUDENT TO APPLY IT TO SOLVE SIMPLE EXERCISES, EVEN IN AN ANALYTICAL AND PHYSICAL-MATHEMATICAL FRAMEWORK. THE EXAM WILL CONSIST OF AN ORAL INTERVIEW. 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.

Verification of learning

THE EXAM WILL CONSIST OF AN ORAL INTERVIEW AIMED TO VERIFY THE LEARNING OF THE THEORY ILLUSTRATED DURING THE COURSE, THE UNDERSTANDING OF ITS ROLE IN THE CONTEMPORARY MATHEMATICS LANDSCAPE, AND THE ABILITY OF THE STUDENT TO APPLY IT TO SOLVE SIMPLE EXERCISES, EVEN IN AN ANALYTICAL AND PHYSICAL-MATHEMATICAL FRAMEWORK.
THE EXAM WILL CONSIST OF AN ORAL INTERVIEW.

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.

	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

Lessons Timetable

BETA VERSION Data source ESSE3 [Ultima Sincronizzazione: 2023-06-01]