Course detail

Differential Geometry

FSI-SDGAcad. year: 2019/2020

The classical differential geometry of curves and surfaces: contact of curves, Frenet formulas, osculating curves, contact of surfaces, the first and the second fundamental form, asymptotic curves, Gauss curvature, ruled surfaces, the intrinsic geometry of a surface. Elements of Tensor Calculus.

Learning outcomes of the course unit

Students will be made familiar with classical differential geometry of curves and surfaces. They will be able to apply this theory in various engineering tasks.


Linear algebra, analytic geometry, differential and integral calculus of functions of one and several variables.


Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

M. Doupovec : Diferenciální geometrie a tenzorový počet (skriptum VUT) (CS)
M. A. Akivis, V. V. Goldberg: An Introduction to Linear Algebra and Tensors, Dover Publications, New York, 1972 (EN)
Manfredo P. do Carmo: Differential Geometry of Curves and Surfaces (Prentice Hall, Inc. 1976) (EN)
I. Kolář, L. Pospíšilová: Diferenciální geometrie křivek a ploch, elektronické skriptum MU (CS)
Boček L.: Tenzorový počet (SNTL Praha) (CS)
A. Pressley: Elementary Differential Geometry, Springer- Verlag, 2012 (EN)
M. Umehara, K. Yamada: Differential Geometry of Curves and Surfaces, World Scientific, 2015 (EN)
M. P. Do Carmo: Differential Geometry of Curves and Surfaces, Prentice Hall, 1976 (EN)
K. Tapp: Differential Geometry of Curves and Surfaces, Springer-Verlag, 2016 (EN)

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.

Assesment methods and criteria linked to learning outcomes

Active attendance at the seminars and written test.
In a 120-minute written test, students have to solve assigned problems.

Language of instruction


Work placements

Not applicable.


The course aims to acquaint the students with the basics of classical differential geometry of curves and surfaces. Another goal of the course is to develop the students' logical thinking.

Specification of controlled education, way of implementation and compensation for absences

Attendance at lectures is recommended, attendance at seminars is required. The lessons are planned on the basis of a weekly schedule. The way of compensation for an absence is fully at the discretion of the teacher.

Classification of course in study plans

  • Programme B3A-P Bachelor's

    branch B-MAI , 2. year of study, summer semester, 4 credits, compulsory

Type of course unit



26 hours, optionally

Teacher / Lecturer


Week 1: The notion of a curve.
Week 2: The contact of curves.
Week 3: Frenet formulas of a plane curve.
Week 4: Osculating curves.
Week 5: Frenet formulas of a space curve.
Week 6. The notion of a surface.
Week 7: The contact of surfaces.
Week 8: The first fundamental form.
Week 9: The second fundamental form.
Week 10: Asymptotic curves.
Week 11: The Gauss curvature.
Week 12: Ruled surfaces.
Week 13: The intrinsic geometry of a surface.


13 hours, compulsory

Teacher / Lecturer


Seminars related to the lectures given in the previous week.

E-learning texts

Miroslav Doupovec: Diferenciální geometrie a tenzorový počet, skriptum VUT, 1999. (cs)