Introduction to Curves and Surfaces 2

Course guarantor:

Ladislav Hlavatý

Lecturer:

Ladislav Hlavatý

Tutor:

Ladislav Hlavatý

Supervisor:

Department of Physics

Synopsis:

The lecture extends the course 02UKP1. The properties of the first fundamental form are briefly summarized. The concept of the second fundamental form is introduced, leading to the mean and Gaussian curvature. Finally, the usual concepts of Riemann geometry are introduced.

Requirements:

Syllabus of lectures:

Outline of the lecture:

1. First fundamental form – review

2. Second fundamental form

3. Mean and Gaussian curvature of a surface

4. Gauss-Weingarten‘s equations

5. Christoffel‘s symbols

6. Codazzi’s equation

7. Riemann‘s and Ricci’s tensor of curvature

8. Gauss theorema egregium

Outline of the exercises:

1.Metrix tensor of a sphere and torus.

2.Evaluation of the mean and Gaussian curvature with the second fundamental

3.Surfaces with zero mean and zero Gaussian curvature

4.Christoffel’s symbols for a plane and a sphere

5. Riemann’s tensor for a plane and a sphere

Syllabus of tutorials:

Outline of the exercises:

1.Metrix tensor of a sphere and torus.

2.Evaluation of the mean and Gaussian curvature with the second fundamental

3.Surfaces with zero mean and zero Gaussian curvature

4.Christoffel’s symbols for a plane and a sphere

5. Riemann’s tensor for a plane and a sphere

Study Objective:

Knowledge:

To provide the simplest examples of manifolds and their properties.

Acquired skills:

Solve mathematical problems defined on manifolds.

Study materials:

Key references:

[1] L. Hlavatý, Úvod do křivek a ploch (in Czech)

www.fjfi.cvut.cz > katedra fyziky > studentský servis > Doprovod přednášek > Úvod do křivek a ploch

Recommended references:

[2] B. Hostinský, Diferenciální geometrie křivek a ploch, Přírodovědecké nakladatelství v Praze, 1949 (in Czech)

[3] W. Kuehnel, Diferential Geometry, AMS2006

[4] T. Banchoff, S Lovett , Diferential Geometry of Curves and Surfaces, CRC Press 2016

Note:

Time-table for winter semester 2024/2025:

Time-table is not available yet

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans:

• Aplikovaná algebra a analýza (elective course)
• Fyzikální inženýrství - Fyzika plazmatu a termojaderné fúze (elective course)
• Jaderná a částicová fyzika (elective course)
• Matematické inženýrství - Matematická fyzika (elective course)
• Nuclear and Particle Physics (elective course)
• Mathematical Engineering - Mathematical Physics (elective course)
• Physical Engineering - Plasma Physics and Thermonuclear Fusion (elective course)