* 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 n-dimensional oriented surfaces in R(n+1). By viewing such surfaces as level sets of 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. What is differential geometry?
2. Graphs and Level Sets.
3. Vector Fields. The Tangent Space.
4. Surfaces. Vector Fields on Surfaces. Lagrange multiplier.
5. Orientation. The Gauss Map
7. Parallel Transport. Covariant Derivative.
8. The Weingarten Map.
9. Normal Curvature .
10. Curvature of Plane Curves. Frenet-Serret Formulas.
11. Arc Length and Global Parametrization.
12. Differential 1-forms and Line Integrals.
13. Gauss-Kronecker Curvature. Mean Curvature.
14. The Second Fundamental Form.
15. Convex Surfaces. Elements of Critical Point Theory.
- J. A. Thorpe: Elementary Topics in Differential Geometry, Undergraduate Texts in Mathematics
- M. P. do Carmo: Differential Geometry of Courves and Surfaces
- J. Oprea: Differential Geometry and Its Applications, 2nd edition
- A. Pressley: Elementary Differential , Undergraduate Mathematics Series
- W. Kuhnel: Differential Geometry: Curves - Surfaces - Manifolds
- M. Spivak: A Comprehensive Introduction to Differential Geometry, Vols. I-V
- 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