* Load is given in academic hour (1 academic hour = 45 minutes)
COURSE AIMS AND OBJECTIVES:
The aim of the Course is to develope the geometry of (parametrized ) n-dimensional oriented surfaces in R(n+k). By viewing such surfaces as level sets of R^k -valued smooth functions , the global ideas can be introduced early without the need for preliminary development of sophisticated machinery. The calculus of vector fields is used as the primary tool in developing the theory. Coordinate patches are introduced only after preliminary discussions of geodesics, parallel transport, curvature, and convexity. Differential 1-forms are introduced only as needed for use in integration.
COURSE DESCRIPTION AND SYLLABUS:
1. Parametrized n-surfaces in (n+k)-dimensional euclidean space.TheTangent and the Normal space . Examples.
2. n-surfaces in (n+k)-dimensional euclidean space . TheTangent and the Normal space . Examples.
3. Local Equivalence of Surfaces and Parametrized Surfaces. Manifolds . Projective spaces, Stiefel and Grassman manifolds.
4. Focal Points.
5. Area (Volume ) of parametrized surfaces.
6. Differential k-forms. Volume form.
7. Partition of Unity and Volume of Surfaces.
8. Minimal Surfaces.
9. The Exponential Map. Geodesic coordinates.
10. Surfaces with Boundary.
11. Exterior Derivative and Stokes Theorem.
12. The Gauss-Bonnet Theorem.
13. Rigid Motions and Congruence.
15. Riemannian Metrics. Models of Geometries.
- J. A. Thorpe: Elementary Topics in Differential Geometry, Undergraduate Texts in Mathematics
- W. Kuhnel: Differential Geometry: Curves - Surfaces - Manifolds
- J. Oprea: Differential Geometry and Its Applications, 2nd edition
- M. Spivak: A Comprehensive Introduction to Differential Geometry, Vols. I-V
- M. P. do Carmo: Differential Geometry of Courves and Surfaces
- A. Pressley: Elementary Differential Geometry, Undergraduate Mathematics Series
- A. Gray: Modern Differential Geometry of Curves and Surfaces, 2nd edition
- D. W. Henderson: Differential Geometry: A Geometric Introduction
- S. - S. Chern, W. H. Chen, K. S. Lan: Lectures on Differential Geometry
- M. Berger: Panoramic View of Riemannian Geometry
Differential geometry 1