# Matematica | GEOMETRY IV

## Matematica GEOMETRY IV

 0512300013 DIPARTIMENTO DI MATEMATICA EQF6 MATHEMATICS 2021/2022

 OBBLIGATORIO YEAR OF COURSE 3 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
 FABRIZIO PUGLIESE T
Objectives
THE COURSE'S MAIN AIM IS TO GIVE BASIC NOTIONS OF: AFFINE GEOMETRY OF QUADRICS; DIFFERENTIAL GEOMETRY OF SUBMANIFOLDS IN EUCLIDEAN SPACES.

- KNOWLEDGE AND UNDERSTANDING: THIS IS AN UNDERGRADUATE COURSE ON "CLASSICAL" DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES IN R^3 AND, MORE GENERALLY, OF SMOOTH SUBMANIFOLDS IN R^N. THE STUDENT LEARN HOW TO APPLY THE PREVIOUSLY LEARNED NOTIONS FROM CALCULUS AND LINEAR ALGEBRA TO THE STUDY OF INFINITESIMAL PROPERTIES OF GEOMETRICAL OBJETS; FURTHERMORE, HE/SHE WILL SEE HOW SUCH INFINITESIMAL PROPERTIES AFFECT THE GLOBAL AND TOPOLOGICAL ASPECTS OF SUBMANIFOLDS (FOR EXAMPLE, THE MINIMUM PROPERTIES OF GEODESICS AND ZERO MEAN CURVATURE SURFACES, GAUSS-BONNET THEOREM, ETC.). FURTHERMORE, THE BASICS OF AFINE GEOMETRY OF QUADRICS WILL BE TAUGHT, SO AS TO ENABLE THE STUDENTS TO RECOGNIZE THE AFFINE TYPE OF A QUADRIC FROM ITS EQUATION IN GENERIC AFFINE COORIDNATES, AND DETERMINE ITS CENTER, AXES, AND SO ON.

- APPLYING KNOWLEDGE AND UNDERSTANDING: THE AIM IS TO ENABLE STUDENTS TO APPLY THE THEORETICAL NOTIONS AND COMPUTATIONAL TOOLS THEY WILL LEARN. TO THIS AIM, MANY LECTURES WILL BE DEVOTED TO PROBLEM SESSIONS (ABOUT 1/3 OF THE TOTAL AMOUNT OF TIME)
Prerequisites
THE NECESSARY (BUT NOT MANDATORY) PREREQUISITE IS THE KNOWLEDGE OF BASIC NOTIONS IN LINEAR ALGEBRA, ANALYTIC GEOMETRY AND CALCULUS USUALLY GIVEN IN THE STANDARD UNDERGRADUATE COURSES.
Contents
THE COURSE CONSISTS OF A SINGLE 48 HOURS UNIT. BELOW YOU FIND THE DETAILED PROGRAM, WITH THE NUMBER OF HOURS PLANNED FOR EACH SUBJECT.

1. AFFINE QUADRICS. (6 HOURS OF LECTURES + 3 OF PROBLEM SESSIONS)

QUADRICS IN AN AFFINE SPACE; CENTERS OF SYMMETRY AND SINGULAR POINTS OF A QUADRIC; TANGENT HYPERPLANES; HYPERBOLIC, ELLIPTIC AND PARABOLIC POINTS OF REAL AFFINE QUADRICS; ACTION OF THE AFFINE GROUP ON A QUADRIC. QUADRICS IN EUCLIDEAN SPACES. ORTHOGONAL SYMMETRY AXES, EUCLIDEAN INVARIANTS; EUCLIDEAN CLASSIFICATION AND CANONICAL EQUATIONS OF QUADRICS. PROJECTIVE CLOSURE OF AN AFFINE QUADRIC. POLARITY WITH RESPECT TO A NON DEGENERATE QUADRIC.

2. DIFFERENTIAL GEOMETRY OF CURVES. (LECTURES: 8 HOURS, PROBLEM SESSIONS: 4 HOURS)

DIFFERENTIABLE CURVES IN R^N; RIPARAMETRIZATIONS; ARC LENGTH AND NATURAL PARAMETERS. OBSCULATING SPACES, FRENET FRAMES AND FRENET EQUATIONS; EXISTENCE AND UNIQUENESS UP TO ISOMETRIES OF CURVES WITH GIVEN CURVATURES. CURVES IN R^2 AND R^3: FRENET EQUATIONS WITH ANY PARAMETER; GEOMETRIC MEANING OF CURVATURE AND TORSION. 1-PARAMETER GROUPS OF LINEAR TRANSFORMATIONS, EXPONENTIAL OF A MATRIX; KINEMATICS OF RIGID BODIES.

3. BASIC NOTIONS OF DIFFERENTIAL CALCULUS ON SUBMANIFOLDS. (LECTURES: 4 HOURS, PROBLEM SESSIONS: 2 HOURS)

LOCAL INVERTIBILITY, IMPLICIT FUNCTION AND RANK THEOREMS; DIFFERENTIABLE SUBMANIFOLDS OF R^N AND THEIR ATLASES. VECTOR FIELDS ON A MANIFOLD AND THEIR FLOWS; LIE DERIVATIVE. SOME HINTS ON DIFFERENTIAL FORMS.

4. DIFFERENTIAL GEOMETRY OF SURFACES. (LECTURES: 14 HOURS, PROBLEM SESSIONS: 7 HOURS)

FIRST FUNDAMENTAL FORM AND INTRINSIC GEOMETRY OF A SURFACE. COVARIANT DERIVATIVE. SHAPE OPERATOR AND SECOND FUNDAMENTAL FORM; NORMAL CURVATURES, PRINCIPAL CURVATURES AND DIRECTIONS, TOTAL AND MEAN CURVATURES; HYPERBOLIC, ELLIPTIC AN PARABOLIC POINTS; RULED AND DEVELOPABLE SURFACES; ROTATION SURFACES. CURVATURE TENSOR AND THEOREMA EGREGIUM. PARALLEL TRANSPORT; GEODESIC CURVATURE; GEODESICS AND THEIR VARIATIONAL PROPERTIES. DIFFERENTIAL OPERATORS ON SUBMANIFOLDS (GRADIENT, DIVERGENCE, LAPLACIAN). GAUSS-BONNET THEOREM.
Teaching Methods
THE COURSE IS DIVIDED INTO LECTURES (32 HOURS) AND PROBLEM SESSIONS (16 HOURS). DURING THE LECTURES THE RESULTS OF DIFFERENTIAL GEOMETRY AND THEORY OF QUADRICS LISTED IN SECTION "CONTENTS OF THE COURSE" WILL BE EXPOSED; IN THE PROBLEM SESSIONS IT WILL BE TAUGHT HOW TO APPLY SUCH RESULTS TO CONCRETE PROBLEMS; FURTHERMORE, THE PROOF OF SOME THEOREMS WILL BE SPLIT INTO AS A SEQUENCE OF EXERCISES, WHOSE SOLUTION WILL TRAIN THE STUDENT TO REASON AUTONOMOUSLY AND MAKE HIS/HER OWN PROOFS. PART OF THE PROBLEMS WILL BE DISCUSSED AND SOLVED IN THE CLASSROOM, WHILE OTHERS WILL BE ASSIGNED AS HOMEWORK, SO THAT STUDENTS CAN DEVELOP THEIR ABILITY TO SOLVE PROBLEMS AUTONOMOUSLY.
Verification of learning
THE EXAM IS ORAL AND APPROXIMATELY 45 MINUTES LONG. THE STUDENTS WILL BE ASKED TO ANSWER TWO QUESTIONS ON THE THEORY AND TO DISCUSS THE SOLUTION OF ONE OF THE PROBLEMS PROPOSED IN CLASSROOM OR AS HOMEWORK; EACH OF THESE THREE "QUIZZES" CORRESPONDS TO ONE OF THE THREE PARTS (QUADRICS, SMOOTH CURVES, SMOOTH SURFACES) OF THE PROGRAM: THE FINAL EXAMINATION MARK WILL BE A WEIGHTED COMBINATION OF THE THREE ANSWERS, WITH THE WEIGHTS BEING APPROXIMATELY PROPORTIONAL TO THE SIZE OF THE CORRESPONDING PART OF THE PROGRAM.
Texts
REFERENCE TEXTS:

- F.PUGLIESE: LECTURE NOTES AND SOLVED PROBLEMS ON QUADRICS AND DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES (DOWNLOADABLE AT HTTPS://DOCENTI.UNISA.IT/004411/RISORSE)

- E. SERNESI, GEOMETRIA 2, BOLLATI BORINGHIERI, 1989, CHAP. VI

- E. VINBERG, A COURSE IN ALGEBRA, AMS 2003, CHAP. VII