1. SylabUZ
2. Wydział Matematyki, Informatyki i Ekonometrii
3. semestr zimowy 2022/2023
4. Mathematics - drugiego stopnia z tyt. magistra
5. Differential Geometry

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Differential Geometry - opis przedmiotu

Cel przedmiotu

To familiarize students with the basics of differential geometry.

 

 

Wymagania wstępne

Zapisz zmiany

Multivariable calculus, linear algebra, topology.

Zakres tematyczny

Lecture

Local theory of curves

1. Curve parameterization,  arclength parameterization (2 hours)
2. Curve length (1 hour)
3. Frenet's frame (1 hour)
4. Frenet's formulas (1 hour)
5. Curvature and torsion of the curve (1 hour)
6. Characterization of curves using curvature and torsion (2 hours)
7. Canonical form of the curve (1 hour)

Global curve theory

1. Fundamental theorem of curve theory
2. Crofton's formula (1 hour)
3. Fenchel's theorem (2 hours)
4. Schur's theorem (1 hour)
5. Four-vertex theorem (1 hour)
6. Isoperimetric inequality (1 hour)

Local surface theory

1. Surface parameterizations (2 hours)
2. First fundamental form (1 hour)
3. Surface area (1 hour)
4. Shape operator (2 hours)
5. Second fundamental form (1 hour)
6. Gaussian curvature and mean curvature (1 hour)
7. Theorema Egregium (2 hours)

Global surface theory

1. Liebmann's theorem (1 hour)
2. Fundamental theorem of surface theory (1 hour)

Minimal surfaces

1. Examples of minimal surfaces (1 hour)
2. Soap bubbles as a physical model of minimal surfaces (2 hours)

 

Exercises

Local theory of curves

1. Determining the arclength of curves. (3 hours)
2.   Arclength parameterizations (1 hour)
3. Curve length calculation (1 hour)
4. Determining Frenet's frame  (2 hours)
5. Calculation of curvature and torsion of curves (2 hours)
6. Determining curves based on curvature and torsion (2 hours)

Global curve theory

1. Determining the vertices of curves (1 hour)

Local surface theory

1. Determining surface parameterization (3 hours)
2. Stereographic projection (1 hour)
3. Determining coefficients of the first form (1 hour)
4. Calculating surface area (1 hour)
5. Determining the second form (1 hour)
6. Discussing issues related to the shape operator using surface models made of plastic masses (2 hours)
7. Determining the shape operator matrix (1 hour)
8. Calculation of Gaussian curvature and mean curvature (2 hours)
9. Geodetic (1 hour)

Minimal surfaces

1. Minimal surfaces in art – examples from the Internet (1 hour)
2. Experiments on soap bubbles. (2 hours.)

Colloquium (2 hours)

Metody kształcenia

Conventional lecture with emphasis on joint discussion of the problems discussed. During classes, students solve tasks together (usually given a week in advance). Blackboard discussions with multiple students are preferred. Constant access to the Internet is assumed (all examples, especially graphics, animations).

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

The condition for passing the exercises is a positive grade in the test. It is allowed to present a paper on differential geometry. The topic is to be chosen independently by the student. Papers can be prepared by a group of two or three students. The topic of the paper must be approved by all students and the instructor. The exam is in written form with the possibility of discussion of solutions between the examiner and the student being examined. The grade for the course consists of the grade for the exercises (40%) and the grade for the exam (60%). The condition for taking the exam is a positive grade from the exercises. The condition for passing the course is a positive grade in the exam.

Literatura podstawowa

1. T. Shifrin, Differential Geometry: A First Course in Curves and Surfaces, 2007. (www.math.uga.edu/~shifrin/ShifrinDiffGeo.pdf)
2. J. Oprea, Geometria różniczkowa I jej zastosowania,  PWN, Warszawa, 2002.

Literatura uzupełniająca

H. Hopf, Differential Geometry in the Large, Springer, 1983.

Uwagi

Zmodyfikowane przez dr Ewa Sylwestrzak-Maślanka (ostatnia modyfikacja: 27-03-2024 16:43)