# Matematica | GEOMETRY III

## Matematica GEOMETRY III

 0512300009 DIPARTIMENTO DI MATEMATICA MATHEMATICS 2015/2016

 OBBLIGATORIO YEAR OF COURSE 2 YEAR OF DIDACTIC SYSTEM 2010 SECONDO SEMESTRE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/03 6 48 LESSONS COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
 ANNAMARIA MIRANDA T
Objectives
TRAINING PURPOSES
-KNOWLEDGE AND UNDERSTANDING
WHAT IS A GEOMETRY? THE COURSE OF GEOMETRY III HAS BEEN THOUGHT OF AS GIVING AN ANSWER TO THIS NATURAL QUERY BY CONSTRUCTING SEVERAL EXAMPLES OF GEOMETRIES FOLLOWING THE ERLANGEN PROGRAM BY F. KLEIN. FOR GENERATING INTEREST AND CURIOSITY AROUND THEOREMS GENERALLY CONSIDERED HARD, FOR REINTERPRETING THEM IN AN ORIGINAL WAY VERSUS THEIR POPULARIZATION, AN INTRODUCTION TO THE EUCLIDEAN AND HYPERBOLIC GEOMETRIES BOTH AS A “PLAY OF MIRRORS” IS GIVEN. ANOTHER PURPOSE IS TO DEEPEN THE KNOWLEDGE OF THE EUCLIDEAN GEOMETRY BY REFINING THE “EUCLIDEAN EYE” IN PARTICULAR IN DIMENSION TWO AND THREE. ANOTHER TRAINING PURPOSE, WORKING EFFICIENTLY FOR DROPPING EUCLIDEAN PREJUDICES, IS TO INTERPRETER THE SAME REALITY BY USING THE “ AFFINE EYE” OR ALSO THE “AFFINE CONFORMAL EYE”. A NEXT OBJECTIVE IS THE CONSTRUCTION OF MODELS OF PLANE GEOMETRIES OF HYPERBOLIC TYPE, AS THE POINCARÉ PLANE AND POINCARÉ HALF-PLANE, OR OF ELLIPTIC TYPE AS THE SPHERICAL SURFACE AND THE REAL PROJECTIVE PLANE. TO THESE TOPICS THE STUDY OF CONICS WITH RELATIVE PROJECTIVE, AFFINE AND METRIC CLASSIFICATION IS ADDED. ITS MAIN SCOPE IS THE ANALYTIC RECASTING OF SYNTHETIC PROPERTIES BY USING ALREADY ACQUIRED MATHEMATICAL KNOWLEDGE. THE ARGUMENTS, TOOLS AND METHODS ARE TARGETED AT INTEGRATING EXPERIENCE, KNOWLEDGE, LEARNING, CURIOSITY AND SKILLS. ASSIGNMENTS, HINTS, SUGGESTIONS ARE GIVEN TO IMPROVE APPLYING ABILITY AND INVENTION IN DEMONSTRATION.
AT THE END OF THE COURSE THE STUDENTS SHOULD HAVE
• ACQUIRED THE BASIC INTRODUCED CONCEPTS, AS , FOR EXAMPLE, THE EUCLIDEAN GEOMETRY AND NON-EUCLIDEAN GEOMETRIES, PROJECTIVE AND HYPERBOLIC GEOMETRIES
• UNDERSTOOD THE RELATIONS AMONG THEM JOINTLY WITH THE USED TECHNIQUES TO STATE THEM, AS, FOR EXAMPLE, IN THE CONSTRUCTION OF NON-EUCLIDEAN MODELS
• ACQUIRED A DEEP KNOWLEDGE AND UNDERSTANDING OF THE EUCLIDEAN WORLD BUT ALSO THE ABILITY TO LOOK AT THE THINGS WITH DIFFERENT “ GEOMETRIC EYES”
• TO CONSIDER THE EUCLIDEAN AND HYPERBOLIC GEOMETRY AS “ A MIRROR PLAY “
-APPLYING KNOWLEDGE AND UNDERSTANDING
PROBLEMS ARE SUGGESTED AND TYPICAL MATHEMATICAL APPROACHES ARE CONSIDERED TO IMPROVE THE APPLYING ABILITY AND CAPACITY OF INVENTION IN DEMONSTRATION. AT THE END OF THE COURSE THE STUDENTS MUST BE ABLE :
• TO USE EFFICIENTLY THE PROPOSED TECHNIQUES BY THEIR APPLICATION IN CONSTRUCTING SIGNIFICANT EXAMPLES AND IN SOLVING PROBLEMS AND EXERCISES
• TO ANALYZE PROBLEMS, EXPLAIN CONCEPTS AND MAKE PROOFS.
-MAKING JUDGEMENTS
THE STUDENTS SHOULD ENRICH THEIR MAKING JUDGEMENTS BY DEEPENING SUGGESTED MATHEMATICAL ARGUMENTS AND TOOLS. BY THEMSELVES, THEY MUST JOIN UP THE DISCIPLINE WITH ALREADY ACQUIRED KNOWLEDGE, IN PARTICULAR WITH THE COMPUTER-SCIENCE INSTRUMENTS.
-COMMUNICATION SKILLS
ONE OF THE PURPOSE OF THE COURSE IS TO MAKE THE STUDENTS ABLE TO EXPRESS THEIR MINDING IN A CLEAR AND RIGOROUS WAY AND TO TELL MATHS IN A POPULAR WAY.
-LEARNING SKILLS
BY THEMSELVES, THE STUDENTS SHOULD BE ABLE TO UNDERSTAND FURTHER AND MORE DIFFICULT GEOMETRICAL CONCEPTS, TO RECOGNIZE TO DIFFERENT GEOMETRIES THE SAME DIGNITY, TO DISCRIMINATE THEIR RELATIVE INVARIANTS AND, FINALLY, TO INTEGRATE EXPERIENCE, KNOWLEDGE AND LEARNING RELATED TO THE COURSE OF GEOMETRY III.

Prerequisites
FINITE-DIMENSIONAL VECTOR SPACES. LINEAR EQUATIONS. LINEAR MAPS AND MATRICES. BILINEAR AND QUADRATIC FORMS.
DIAGONALIZATION.
Contents
EUCLIDEAN SPACES AND THEIR VECTOR, AFFINE AND METRIC STRUCTURE. LINEAR MAPS AND MATRICES: THE GENERAL LINEAR GROUPS. ORTHOGONAL TRANSFORMATIONS: ORTHOGONAL GROUPS AND SPECIAL ORTHOGONAL GROUPS.CLASSIFICATION AND DECOMPOSITION OF ORTHOGONAL TRANSFORMATIONS IN LOW DIMENSIONS: EULER THEOREM AND CARTAN-DIEUDONNÉ THEOREM. THE EUCLIDEAN ISOMETRIES AND EUCLIDEAN GEOMETRY: THE FUNDAMENTAL DECOMPOSITION THEOREM. CHASLES THEOREM. THE CARTAN-DIEUDONNÉ THEOREM FOR ISOMETRIES: EUCLIDEAN GEOMETRY AS A PLAY OF MIRRORS.
ELEMENTARY GEOMETRICAL FIGURES AND THEIR SYMMETRIES. SIMILARITIES AND THEIR DECOMPOSITION. PROJECTIVE REAL PLANE:CONSTRUCTION OF MODELS OF THE REAL PROJECTIVE PLANE. CONICS AND THEIR PROJECTIVE, AFFINE, METRIC CLASSIFICATION. AN INTRODUCTION TO HYPERBOLIC GEOMETRY: THE POINCARÉ PLANE AND HALF-PLANE. CROSS-RATIO. HYPERBOLIC DISTANCE. BETWEENNESS. CIRCULAR INVERSIONS. HYPERBOLIC GEOMETRY AS A PLAY OF MIRRORS.

Teaching Methods
TO REDUCE COMPLEXITY TO SIMPLICITY AND CONTEMPORANEOUSLY DISPLAY AND MAKE GRADUALLY ACCESSIBILE STANDARD DEMONSTRATIVE TECHNIQUES. FURTHER, TO MAKE EASY TO UNDERSTAND AND APPRECIATE HOW FUNDAMENTALS THEOREMS PERFORM SIMPLICITY ; FOR INSTANCE, PROVING THAT EUCLIDEAN ISOMETRIES CAN BE GENERATED BY ELEMENTARY TRANSFORMATIONS AS TRANSLATIONS AND ORTHOGONAL TRANSFORMATIONS, OR DECOMPOSING THEM IN REFLECTIONS, SO STARTING WITH A “PLAY OF MIRRORS”. TO EXPLORE THE GEOMETRY OF COMPLEX OBJECTS BY OBJECTS WITH A SIMPLER STRUCTURE, FOR EXAMPLE BY EXPLORING CONICS WITH STRAIGHT LINES OR BY PENCIL OF STRAIGHT LINES. TO EXPLAIN THAT THERE IS NO ABSOLUTE GEOMETRY, BUT ONLY GEOMETRIES CLOSER THAN SOME OTHER ONES TO THE WORKING CONTEXT; FOR INSTANCE, CONSTRUCTING CONCRETE MODELS OF NON-EUCLIDEAN GEOMETRY AS THE POINCARÉ PLANE AND SPHERICAL GEOMETRY. THE TOOLS AND METHODS ARE TARGETED AT INTEGRATING EXPERIENCE, KNOWLEDGE, LEARNING, CURIOSITY AND SKILLS. ASSIGNMENTS, HINTS, SUGGESTIONS ARE GIVEN TO IMPROVE APPLYING ABILITY AND INVENTION IN DEMONSTRATION.
Verification of learning
A FINAL WRITTEN AND ORAL EXAMINATION TO VALUE THE KNOWLEDGE OF THE ARGUMENTS TREATED IN THE COURSE, THE LEVEL OF UNDERSTANDING OF PERFORMED MATHEMATICAL APPROACHES, THE COMMUNICATION SKILLS, THE OPENING IN DISCUSSION, THE ORIGINALITY IN ARGUMENTATION AND INVENTION IN DEMONSTRATION.
Texts
1] E. AGAZZI- D. PALLADINO, LE GEOMETRIE NON EUCLIDEE E I FONDAMENTI DELLA GEOMETRIA DAL PUNTO DI VISTA ELEMENTARE, LA SCUOLA, 1998.

[2] D. HILBERT- S. COHN-VOSSEN, GEOMETRIA INTUITIVA, BORINGHIERI, 2000.

[3] E. SERNESI GEOMETRIA 1 BOLLATI-BORINGHIERI 2000.

[4] E. SERNESI GEOMETRIA 2 BOLLATI- BORINGHIERI, 2001.

[5] G. TALLINI STRUTTURE GEOMETRICHE LIGUORI EDITORE.