Lectures: TR 13:00-14:15PM, Room VAN HISE 115
Office Hours: T 14:30-16:00PM in Van Vleck 719
DESCRIPTION:
Differential geometry is a mathematical discipline that uses the techniques of differential calculus and (multi)linear algebra to study problems in geometry.
The theory of plane and space curves and surfaces in the three-dimensional Euclidean space, which will be studied in the course Math 561, was developed during the 18th century and the 19th century and contains some of the most beautiful results in mathematics.
In particular, two gems by C. F. Gauss, the Princeps mathematicorum, will be treated.
The geodetic survey of he Kingdom of Hanover, which required Gauss to spend summers traveling on horseback for a decade, fueled Gauss's interest in differential geometry and topology.
Among other things he came up with the notion of Gaussian curvature and the Theorema Egregium (remarkable theorem), establishing an important property of the notion of curvature. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring angles and distances on the surface. That is, curvature does not depend on how the surface might be embedded in 3-dimensional space.
Gauss-Bonet Theorem is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic).
Math 561 is an excellent introduction to more advanced courses on Riemannian, Finsler, Sympletic, Contact, Complex and Kaehler geometry, to Differential Topology, to Lie Groups, to the mathematical aspects of General Relativity Theory, to Lagrangian and Hamiltonian Mechanics...
TOPICS:
-Geometry of curves, parametrized curves, regular curves, arc length, curvature and torsion, Frenet frame, fundamental theorem of the local theory, some global properties.
-Geometry of surfaces, regular surfaces, parametrized surfaces, tangent space, normal vectors, first and second fundamental forms, notions of curvature, fundamental equations and theorem of surfaces, Egregium theorem, Gauss-Bonet.
READING SOURCES:
- Differential Geometry of Curves and Surfaces, Thomas F. Banchoff, S
CHAPMAN AND HALL/CRC, ISBN 9781482247343 (Required textbook)
- Differential Geometry of Curves and Surfaces, M. Do Carmo, Pearson, ISBN 9780132125895
- Differential Geometry Curves - Surfaces - Manifolds, W. Kuehnel, AMS, ISBN 9781470423209
-Curves and Surfaces, AMS-RSME, Graduate Studies in Mathematics 69, 2005, ISBN 9780821838150
- Elementary Differential Geometry, A. Pressley, Springer, ISBN 978-1-84882-890-2
GRADE:
Homework and class participation: 25%, Midterm: 25%, Final Exam: 50%.
FINAL EXAM:
Thursday, May 12, 17:05-19:05, Van Hise 115.
HOMEWORK:
It will be assigned regularly and will be the main source of problems appearing in the exams.
First homework assignment, due on March 1st (in class).
Second homework assignment, due on April, 5th (in class).
Third homework assignment, due on May 5th (in class).