1. komponenta

Lecture type	Total
Lectures	45

* 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.
14. Isometries.
15. Riemannian Metrics. Models of Geometries.

1. Elementary Topics in Differential Geometry, Undergraduate Texts in Mathematics, J. A. Thorpe, Springer Verlag, 1994.
2. Differential Geometry: Curves - Surfaces - Manifolds, W. Kuhnel, American Mathematical Society, 2002.
3. Differential Geometry and Its Applications, 2nd edition, J. Oprea, Prentice Hall, 2003.
4. A Comprehensive Introduction to Differential Geometry, Vols. I-V, M. Spivak, Publish or Perish, Boston, 1970.
5. Differential Geometry of Courves and Surfaces, M. P. do Carmo, Prentice Hall, 1976.
6. Elementary Differential Geometry, Undergraduate Mathematics Series, A. Pressley, Springer Verlag, 2001.
7. Modern Differential Geometry of Curves and Surfaces, 2nd edition, A. Gray, CRC Press, 1997.
8. Differential Geometry: A Geometric Introduction, D. W. Henderson, Prentice Hall, 1998.
9. Lectures on Differential Geometry, S. - S. Chern, W. H. Chen, K. S. Lan, World Scientific Publishing, 1999.
10. Panoramic View of Riemannian Geometry, M. Berger, Springer Verlag, 2003.

Enrollment :
Attended : Differential geometry 1

Examination :
Passed : Differential geometry 1

Mandatory course - Regular study - Theoretical Mathematics