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Differential geometry 2

Code: 92902
ECTS: 5.0
Lecturers in charge: izv. prof. dr. sc. Ilja Gogić - Lectures
English level:

1,0,0

All teaching activities will be held in Croatian. However, foreign students in mixed groups will have the opportunity to attend additional office hours with the lecturer and teaching assistants in English to help master the course materials. Additionally, the lecturer will refer foreign students to the corresponding literature in English, as well as give them the possibility of taking the associated exams in English.
Load:

1. komponenta

Lecture typeTotal
Lectures 45
* Load is given in academic hour (1 academic hour = 45 minutes)
Description:
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.
Literature:
  1. J. A. Thorpe: Elementary Topics in Differential Geometry, Undergraduate Texts in Mathematics
  2. W. Kuhnel: Differential Geometry: Curves - Surfaces - Manifolds
  3. J. Oprea: Differential Geometry and Its Applications, 2nd edition
  4. M. Spivak: A Comprehensive Introduction to Differential Geometry, Vols. I-V
  5. M. P. do Carmo: Differential Geometry of Courves and Surfaces
  6. A. Pressley: Elementary Differential Geometry, Undergraduate Mathematics Series
  7. A. Gray: Modern Differential Geometry of Curves and Surfaces, 2nd edition
  8. D. W. Henderson: Differential Geometry: A Geometric Introduction
  9. S. - S. Chern, W. H. Chen, K. S. Lan: Lectures on Differential Geometry
  10. M. Berger: Panoramic View of Riemannian Geometry
Prerequisit for:
Enrollment :
Passed : Differential geometry 1
2. semester
Mandatory course - Regular study - Theoretical Mathematics
Consultations schedule: