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Subject: GEOMETRY OF SURFACES (A.A. 2021/2022)

master degree course in MATHEMATICS

Course year 2
Teaching units Unit Geometria delle superfici
Advanced Theoretical Studies (lesson)
  • TAF: Compulsory subjects, characteristic of the class SSD: MAT/03 CFU: 6
Teachers: Alberto CAVICCHIOLI
Exam type oral
Evaluation final vote
Teaching language Italiano
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The aim of the course is to give the basic knowledge, the operative methods and techniques of the Differential Geometry of Surfaces immersed in the Euclidean 3-space and study in details the Theory of principal differentiable surfaces.

Admission requirements

The students should have the basic knowledge given by the courses of the first triennium of their studies.

Course contents

Differential geometry of surfaces. Differentiable surfaces in Euclidean space. Differentiable functions defined on a surface. Mappings of Surfaces. Diffeomorphisms. Vector fields. Arcs and Curves on a surface. Oriented atlas. Tangent plane. Normal unitary vector. First fundamental quadratic form of a surface. Metric on a surface. Area of a surface. Sectional curvature and normal curvature of a curve on a surface. Second fundamental quadratic form of a surface. Asymptotic tangents. Elliptic, parabolic and hyperbolic points. Asymptotic curves. Multiple points of a surface. Tangent cone to a surface. Principal curvatures. Gaussian Curvature and Mean Curvature.. Isometries. Local isometries. Geodesics on a surface. The Gauss theorems. The Gauss-Bonnet theorem.
Special differentiable surfaces. Surfaces of Revolution. Cylinders. Ruled Surfaces. Catenoids. Translation Surfaces. Conoids. Elicoids. Algebraic Surfaces. Rational Surfaces. Monoids. Quadrics. Minimal Surfaces. Surfaces of Constant Curvature. Introduction to differential nonj-euclidean geometries. Exercises on surfaces and graphical representations of paths and surfaces.

Teaching methods

Frontal lectures which include theory and exercizes. Textbooks written by the professor of the courses. Explanations during the receivement for every week.

Assessment methods

To verify the student profit there will be an oral examination which includes questions of theory and exercizes in the presence or by a written examination on line on PC in Dolly taking 60 minutes with random questions (exercizes + theory) according to the evolution of the COVID19 pandemic problem. In the second case examinations are under video surveillance of the professor.

Learning outcomes

Through lectures and individual study, knowledge and understanding of:
Differentiable surfaces of the Euclidean 3-space: Immersions and embeddings. Vector fields along a surface. Velocity vectors. Singular and regular points. Tangent plane and normal line. Directional derivatives. Differential forms. First fundamental quadratic form of a surface. Normal curvature. Sectional Curvature. Gaussian curvature of a surface. Theorem of Meusnier. Principal curvatures. Theorem of Eulero. Area of a compact surface. Classification of the points of a surface. Geodesics. Special classes of surfaces. Remarkable algebraic surfaces. Exercises on special surfaces and graphical representation of them. Through classroom exercises, the support activities and individual work, ability to discuss and to solve every problems regarding the geometry of differentiable surfaces in the Euclidean space and the geometry of special classes of surfaces.

Attitude towards a methodological approach that leads to verify the claims through rigorous arguments and methods presented.
Ability to self-assessment of their skills and abilities.

Ability to deal with accurately and coherent a dialectic, arguing with precision

Acquisition of mathematical knowledge as its assets, that can use at any time of the cultural path.
Attitude towards a methodological approach that leads to an improvement in the method of study and in the learning ability.


1)A. Cavicchioli-M. Meschiari, Lezioni di Geometria: seconda parte, Pitagora Editrice, Bologna, 1996

2) A. Cavicchioli-F. Hegenbarth, Lezioni di Topologia Algebrica e Differenziale, Pitagora Editrice, Bologna, 1997.

3)B. Ruini-F. Spaggiari, Esercizi di Geometria, Pitagora Editrice, Bologna, 2002

4) B. O'Neill, Elementary Differential Geometry, Academic Press, New York-San Francisco-London.

5)R. Caddeo-A. Gray, Curve e superfici, CUEC 2002.

6) Abate, M., Tovena, F., Curves and Surfaces, Springer Verlag, 2012.