Introduction to differential geometry

2022/2023
Programme:
Mathematics, First Cycle
Year:
3 year
Semester:
first
Kind:
optional
Group:
B
ECTS:
5
Language:
slovenian
Hours per week – 1. semester:
Lectures
2
Seminar
0
Tutorial
2
Lab
0
Prerequisites

Completed courses Analysis 2a and Analysis 2b.

Content (Syllabus outline)

Introduction: Definition of a curve and of a surface with some examples. Abstract curves and surfaces. Curves and surfaces, embedded in three-dimensional real space. Frenet frame. Frenet formulae. Charts, atlases, orientability, revision of the Euler characteristic and genus. Smooth surfaces.
Metric: Isometry of surfaces. The concept of the metric. Isometry.
Curvature: Second fundamental form of an imbedded surface. Normal and geodesic curvatures of a curve on a surface. Gaussian curvature. Various descriptions and interpretations of the Gaussian curvature. Mean curvature and minimal surfaces. Geodesic curves. Geodesic coordinate system.
Gauss' Theorema Egregium: Gaussian curvature as an isometric invariant. Classification of surfaces with constant curvature.
Gauss-Bonnet Theorem: The local and the global versions of this theorem. Relation between the Euler characteristic (a topological invariant) and the Gaussian curvature (geometric invariant). Algebraic number of stationary points of a vector field on a closed surface. The alternating sum of minima, saddles and maxima of a smooth function on a surface.

Readings

A. Pressley, Elementary Differential Geometry, Springer, Berlin, 2010
Vidav: Diferencialna geometrija, DMFA-založništvo, Ljubljana, 1989.
M. P. do Carmo: Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, 1976.
D. W. Henderson: Differential Geometry : A Geometric Introduction, Prentice Hall, Upper Saddle River, 1997.
I. M. Singer, J. A. Thorpe: Lecture Notes on Elementary Topology and Geometry, Springer, New York-Heidelberg, 1976.

Objectives and competences

Students get acquainted with the elementary notions of differential geometry, namely the metric, the curvature and the geodesic curves. These notions are treated on surfaces which are the simplest nontrivial context for the study of differential geometric constructions. The constructions presented are mainly extrinsic, but nevertheless, the students get acquainted with the difference between the extrinsic and the intrinsic. The relationship between topology and geometry is established by means of the Gauss-Bonnet theorem.

Intended learning outcomes

Knowledge and understanding: Understanding of the fundamental notions of differential geometry: the metric, the curvature and the geodesic curves. Understanding of the difference between topological and geometric properties of spaces. Ability of computational treatment of the notions mentioned above.
Application: Some applications of differential geometry in mechanics and other branches of physics. Use of geometry in the topological study of surfaces. Use of differential geometry in solving certain differential equations.
Reflection: Understanding of the theory from the applications.
Transferable skills: Proficiency in certain chapters of Analysis 1, Analysis 2a and 2b and Linear algebra is needed for successful study of differential geometry. Students learn how to use previously acquired knowledge in the study of new subjects.
Skills related to the use of literature in different languages.

Learning and teaching methods

Lectures, exercises, homework, consultations

Assessment

2 midterm exams instead of written exam, written exam
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

SAKSIDA, Pavle. On zero-curvature condition and Fourier analysis. Journal of physics. A, Mathematical and theoretical, ISSN 1751-8113, 2011, vol. 44, no. 8, 085203 (19 str.). [COBISS-SI-ID 15909465]
SAKSIDA, Pavle. Nahm's equations and generalizations Neumann system. Proceedings of the London Mathematical Society, ISSN 0024-6115, 1999, let. 78, št. 3, str. 701-720. [COBISS-SI-ID 8853849]
SAKSIDA, Pavle. Neumann system, spherical pendulum and magnetic fields. Journal of physics. A, Mathematical and general, ISSN 0305-4470, 2002, vol. 35, no. 25, str. 5237-5253. [COBISS-SI-ID 11920217]
OWENS, Brendan, STRLE, Sašo. Rational homology spheres and the four-ball genus of knots. Advances in mathematics, ISSN 0001-8708, 2006, vol. 200, iss. 1, str. 196-216. [COBISS-SI-ID 13875033]
STRLE, Sašo. Bounds on genus and geometric intersections from cylindrical end moduli spaces. Journal of differential geometry, ISSN 0022-040X, 2003, vol. 65, no. 3, str. 469-511. [COBISS-SI-ID 13135193]
STEFANOVSKA, Aneta, STRLE, Sašo, KROŠELJ, Peter. On the overestimation of the correlation dimension. Physics letters. Section A, ISSN 0375-9601. [Print ed.], 1997, vol. 235, str. 24-30. [COBISS-SI-ID 607828]