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### __ MATH 345 - DIFFERENTIAL GEOMETRY I__

### __Fall 2002,2018 __

** Math345 **: __Differential Geometry I. A third year course.__

__ Text Book: __
Differential Geometry of Curves and Surfaces, by
Manfredo P. do Carmo. Prentice-Hall, New Jersey , 1976.

__ Other books :__
Elementary Differential Geometry, by Barret O' Neil. Academic Press,1997.

Elementary Differential Geometry, by Andrew Pressley, Springer Undergraduate
Mathematics Series, 2012.

** **__ Content: __
In this period the first four
chapters of " M. Do Carmo" will be covered. See the
"Course Syllabus" below

** **__ Course Schedule: __
Monday 15.40-17.30 , SA-Z02
Thursday 13.40-15.30 , SA-Z02

varna2

__ Exams:__
(%20) First Midterm Exam ps file
: October 22 (SA-Z02)
(S (%20) Second Midterm Exam ps file
: November 26 (SA-Z02)
(%40) Final Exam ps file
: January xx (SA-Zxx)
(%20) Homework:
homework and midterm grades

** **__ Subjects Covered __

** 1. September 24 **

Parametrized Curves (1.2)
Regular Curves; Arc Length (1.3)
The Vector Product (1.4)

** 2. October 1 **
The Local Theory of Curves (1.5)
The Local Canonical Form (1.6)
Global Properties of Plane Curves.2

homework assignment Set 1

** 3. October 8 **
Regular Surfaces
Change of Parameters;Difgerential Functions on Surfaces

** 4. October 15 **

The Tangent Plane; the differentail of a map
The First Fundamenatl form

** 5. October 22 **
Orientation of Surfaces
A Characterization of Compact Orientable Surfaces
First Midtrem Exam

Homework Set 2

** 6. October 29 **
The Gauss Map and its Fundamental Properties

Homework Set 3

** 7. November 5 **
The Gauss Map

** 8. November 12**

The Vector fields

** 9. November 19 **

Ruled and Minimal Surfaces

IV th Homework Set

** 10. November 26 **

Differential Forms on Surfaces
Second Midterm Exam

** 11. December 03 **

Isometries

** December 10**

12. The Gauss Theorem

** 13. December 17 **

Parallel Transport

** 14. December 20 **
The exponential Map

__ Course Syllabus __

famous curves

1. Sept.23 (Sections 1.2,1.3) Euclidean Space, Tangent vectors, Directional
Derivatives.

2. Sept.30 (Sections 1.4.1.7) Curves in space, 1-forms, differential Forms,
Mappings.

3. Oct.07 (Sections 2.1- 2.3) Dot product, curves , The Serret Frenet Formulas

4. Oct.14 (Sections 2.4-2.6 Covariant derivative, frame fields, connection
forms ** **

5. Oct.21 (Sections 2.7,2.9) Regular Surfaces, Change of Parameters

6. Oct.28 (Sections 3.1-3.3) Isometries , the Tangent map,
orientation.

7. Nov.4 (Sections 3.5-3.7) Euclidean Geometry, Congruence of curves. (M.Ex.1)

8. Nov.11 (Sections 4.1-4.4 Surfaces in R^3 , Differentiable functions

and Tangent vectors, Differential forms on surfaces.

9. Nov.18 (Sections 4.5-4.6) Mapping of surfaces, integration of forms.

10. Nov.25 (Sections 4.7-4.8) Topological properties of surfaces, Manifolds.

11. Dec.02. (Sections 5.1-5.3) Shape operator, Normal curvature, Gaussian
curvature.

12. Dec.09 (Sections 5.4-5.5) Computational Techniques, the implicit example

13. Dec.16 (Sections 5.6-5.7) Special curves and surfaces, surfaces
of revolution. (M.Ex.2)

14. Dec.23 Integrable Surfaces

15. Dec.30 Integrable surfaces
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### Last update September 2002

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