Matematica | GEOMETRY IV

cod. 0512300013

GEOMETRY IV

	0512300013
	DIPARTIMENTO DI MATEMATICA
	EQF6
	MATHEMATICS
	2020/2021

PERCORSO SHARED STUDY PATHWAY

	OBBLIGATORIO
	YEAR OF COURSE 3
	YEAR OF DIDACTIC SYSTEM 2018
	PRIMO SEMESTRE

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	MAT/03	6	48	LESSONS	COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS

	Objectives
	THE COURSE'S MAIN AIM IS TO GIVE BASIC NOTIONS OF: PROJECTIVE AND 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. - 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.

THE COURSE'S MAIN AIM IS TO GIVE BASIC NOTIONS OF: PROJECTIVE AND 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.

- 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.

	Prerequisites
	THE ONLY NECESSARY PRELIMINARY KNOWLEDGES ARE THE BASIC ONES IN LINEAR ALGEBRA AND CALCULUS USUALLY GIVEN IN THE STANDARD UNDERGRADUATE COURSES.

	Contents
	1. AFFINE QUADRICS. 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. 2. PROJECTIVE QUADRICS. BASIC NOTIONS OF PROJECTIVE GEOMETRY. QUADRICS IN PROJECTIVE SPACES; RANK OF A PROJECTIVE QUADRIC; SINGULAR AND REGULAR POINTS; TANGENT HYPERPLANES; POLARITY; CLASSIFICATION OF REAL AND COMPLEX PROJECTIVE QUADRICS; PROJECTIVE SUBSPACES CONTAINED IN A QUADRIC; RULED QUADRICS. PROJECTIVE EXTENSION OF AN AFFINE QUADRIC. GROUP OF PROJECTIVE TRANSFORMATIONS OF A QUADRIC.. 3. BASIC NOTIONS OF DIFFERENTIAL CALCULUS ON SUBMANIFOLDS. 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 AND EXTERIOR DERIVATIVE. 4. DIFFERENTIAL GEOMETRY OF CURVES. 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. 5. DIFFERENTIAL GEOMETRY OF SURFACES. 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.

Contents

1. AFFINE QUADRICS.

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.

2. PROJECTIVE QUADRICS.

BASIC NOTIONS OF PROJECTIVE GEOMETRY. QUADRICS IN PROJECTIVE SPACES; RANK OF A PROJECTIVE QUADRIC; SINGULAR AND REGULAR POINTS; TANGENT HYPERPLANES; POLARITY; CLASSIFICATION OF REAL AND COMPLEX PROJECTIVE QUADRICS; PROJECTIVE SUBSPACES CONTAINED IN A QUADRIC; RULED QUADRICS. PROJECTIVE EXTENSION OF AN AFFINE QUADRIC. GROUP OF PROJECTIVE TRANSFORMATIONS OF A QUADRIC..

3. BASIC NOTIONS OF DIFFERENTIAL CALCULUS ON SUBMANIFOLDS.

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 AND EXTERIOR DERIVATIVE.

4. DIFFERENTIAL GEOMETRY OF CURVES.

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.

5. DIFFERENTIAL GEOMETRY OF SURFACES.

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
	ABOUT 2/3 OF THE COURSE WILL BE DEVOTED TO LECTURES AND ABOUT 1/3 TO PRACTICAL CLASSES, DURING WHICH VARIUOS EXERCICES AND PROBLEMS WILL BE DONE, WHILE OTHERS WILL BE ASSIGNED AS HOMEWORK, SO THAT STUDENTS CAN DEVELOP THEIR ABILITY TO SOLVE PROBLEMS AUTONOMOUSLY.

	Verification of learning
	THE AIM OF THE FINAL TEST IS TO VERIFY BOTH THE STUDENT'S KNOWLEDGE OF THE THEORY AND HIS/HER ABILITY TO APPLY IT TO THE SOLUTION OF CONCRETE PROBLEMS. THE EXAM, WHICH WILL TAKE PLACE IN ONE SESSION, WILL CONSIST OF: 1. THE ORAL DISCUSSION OF HOMEWORK 2. THE SOLUTION OF A SIMPLE EXERCISE PROPOSED BY THE EXAMINER 3. AN INTERVIEW ON ONE OR TWO ITEMS OF THE PROGRAM, CHOSEN BY THE EXAMINER.

	Texts
	TEXTBOOKS: - LECTURE NOTES BY THE INSTRUCTOR - E. SERNESI, GEOMETRIA 2, BOLLATI BORINGHIERI, 1989 - E. VINBERG, A COURSE IN ALGEBRA, AMS 2003 - M. ABATE, F. TOVENA, CURVE E SUPERFICI, SPRINGER 2006 - BELTRAMETTI ET AL., LEZIONI DI GEOMETRIA ANALITICA E PROIETTIVA, BOLLATI BORINGHIERI 2003