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Second Semester 
Teaching style
Lingua Insegnamento

Informazioni aggiuntive

Course Curriculum CFU Length(h)
[60/64]  MATHEMATICS [64/00 - Ord. 2012]  PERCORSO COMUNE 8 64


Geometry 4 is devoted to the study of "differential geometry" of curves and surfaces. Broadly speaking, “differential geometry” is the use of tools of calculus to study problems in geometry. More specifically, in this course, we will apply tools of multivariable and vector calculus to study geometric properties of curves and surfaces in 3-dimensional space. Whereas topology, the subject matter of Geometria 3, is concerned with properties that are preserved by homeomorphisms (continuous bijections with continuous inverses), geometry is concerned with properties that are preserved by distance-preserving homeomorphisms. Such properties include distances (of course) and angles, as well as other familiar non-topological properties such as lengths of curves, areas, and volumes. And the most important geometric property of all: the curvature.
Specifically, we will cover most or all of the following topics: the theory of curves in the plane and in 3-space, the extrinsic local theory of surfaces in 3-space (looking at a surface from outside), the intrinsic local theory of surfaces (looking at a surface from inside), geodesics, curvature, the Gauss–Bonnet theorem, and the global theory of surfaces. More specifically, the student should acquire the following knowledge and skills.

Know the language of differential geometry of curves and surfaces with particular attention to parameterized curves, parameterized surfaces and intrinsic geometry of surfaces.
Ability to solve standard exercises and new problems, where you need to develop a strategy yourself, or develop a small demonstration similar to those seen in the lectures. In addition, the student must be able to represent parametrizations of remarkable curves and surfaces, know how to calculate the curvatures of curves and surfaces, know how to calculate and solve the equations of geodesics.
Know how to recognize when a logic procedure is correct. Know how to choose the most effective procedure for solving complex exercises.
The student learns to use the modern language of the differential geometry that will allow him to communicate and apply the results of modern geometry correctly.
Ability to learn to solve exercises and complex problems independently. Ability to read and understand an advanced math text even in English.


Linear algebra. Conics and quadrics. Differential Calculation. Rudiments of Topology.


Summaries and complements of linear algebra and analytical geometry. R^n and his group of rigid movements.

Parametric curves. Arc Length . Length of a curve. The signed curvature of a plane curves. The angle of rotation. The fundamental theorem of the curves in the plane. Famous plane curves: cycloid, tractrix, catenary, and Bernoulli's lemniscate. The Frenet's frame and Frenet formulas for curves in space. Calculation of curvature and torsion for curves not parameterized with the arc length. The fundamental theorem of curves in space. Determination of a curve with curvature and torsion assigned: the case where curvature and torsion are constant. Global properties of plane curves.

Parametric surfaces. Regular parametrisations. Injective parametrisations. Differential of an application. Tangent vectors and tangent plane. Regular surfaces of R^n. Metrics on surfaces. The first fundamental form. Orientation of a surface. Orientable surfaces. A geometric definition of area.

Extrinsic geometry of Surface. Definition and properties of the Gauss map. The shape operator. The second fundamental form. Normal curvature. Geometric meaning of Euler's formula. Principal curvature. Gaussian curvature and mean curvature. Gauss map in local coordinates. Ruled surfaces and surfaces of revolution. Minimal Surfaces. Vectors fields on surfaces.

Intrinsic geometry of surfaces. Isometries and conformal maps. The stereographic projection. The formulas of Bianchi, Brioschi and Lie. Theorema Egregium. Compatibility Equations. Parallel transport and covariant derivation. Geodesics on a surface. Geodesic equations. Geodesics on a surface of revolution. Clairaut's relation. Geodetic polar coordinates. Local classification of surfaces with constant Gauss curvature. The Gauss-Bonnet theorem.

Teaching Methods

The use of visualization allows to develop geometric intuition and is a valid aid to mathematical reasoning. The ability of calculations as well as the representation of geometric objects is stimulated by a good number of resolved and proposed exercises. A weekly meeting dedicated to exercises aim to overcome difficulties and improve learning. Mathematica's symbolic computational software will also be used to visualize surfaces and automate most of the standard counts of surface theory in R^3. The course includes 32 lectures of 2 hours each and 16 tutorials of 2 hours each.

Verification of learning

The goal of the exam is to check the level of achievement of the above-mentioned training objectives.
Examination consists of a written test and an oral exam.

The written test (3 hours) consists in a selection of 6 exercises similar to the homework assigned during the course. Each exercise is worth 5 points and the written test is deemed passed if the final score is not less than 18. In this test mainly occur: knowledge and understanding; knowledge and understanding applied; the ability to learn.

The oral exam (usually of 40 minutes) consists of a discussion of at least three arguments. The student must demonstrate that they have understood and assimilated the arguments (knowledge and understanding, knowledge and understanding applied). In addition, students must demonstrate that they are able to explain the concepts and demonstrations learned during the course (Communication Skills). To this end it is necessary that the student, during the oral exam, write on the chalkboard and explain all the steps that it is following to reach the conclusion of an argument (independent judgment). It is advisable to answer exactly the questions without rambling and writing immediately all the necessary details (independent judgment, communication skills).

The final grade after the oral interview is assigned according to the following Docimological Table.

Insufficient: the student proves he did not understand many of the basic constructions of the discipline.

18-24: the student knows almost all the arguments submitted during the examination; demonstrates that has understood and assimilated the arguments sufficiently.

25-28: the student knows all the arguments submitted during the examination; demonstrates that has understood and assimilated well the arguments; the student uses mathematical language correctly.

29-30: the student knows very well all the arguments submitted during the examination; demonstrates that has understood and assimilated very well the arguments; the student uses mathematical language correctly; is able to explain the concepts learned.

30 e lode: the student knows perfectly all the arguments submitted during the examination; demonstrates that has understood and assimilated in depth all the topics; the student uses mathematical language correctly; is able to explain the concepts learned.


L. Woodward, J. Bolton, A First Course in Differential Geometry
Surfaces in Euclidean Space, Cambridge University Press, 2018.

M. P. do Carmo, Differential Geometry of Curves and Surfaces, Dover, 2016.

S. Montiel, A. Ros, Curves and Surfaces, Graduate Studies in Mathematics vol. 69, AMS, 2005

Books for the use of Mathematica
R. Caddeo, A. Gray, Lezioni di geometria differenziale, Volumi I e II, CUEC, Cagliari, 2001 e 2002.

Other books
D. Hilbert, S. Cohn-Vossen, Geometria intuitiva, Boringhieri, Torino, 1972.

More Information

The teacher receives the students, when present in the office, every day of the week both in the morning and in the afternoon, even without appointment. If students wishes to do so, they can request an appointment by email: montaldo@unica.it


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