# Differential Geometry (MATH374) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Differential Geometry MATH374 6. Semester 3 0 0 3 6
Pre-requisite Course(s)
Course Language English Compulsory Departmental Courses Bachelor’s Degree (First Cycle) Face To Face Lecture, Question and Answer, Team/Group. This is classical differential geometry, i.e. differential geometry of curves and surfaces in space. The basic concern is the investigation of geometric properties of curves and surfaces using analysis and linear algebra. The content of the course serves as the intuitive motivation of theory of differentiable manifolds, Riemannian geometry and other modern geometries as well as the indispensable tool for mathematical modeling in classical physics and engineering applications. The students who succeeded in this course; understand geometric properties of curves and surfaces in 3- dimensional space. understand intrinsic geometry, geodesic, curvature and Gauss-Bonnet Theorem. Curves in the plane and space, curvature and torsion, global properties of plane curves, surfaces in space, the First Fundamental Form, curvatures of surfaces, Gaussian curvature and the Gauss Map, geodesics, minimal surfaces, Gauss`s Theorema Egregium, the Gauss-Bonnet Theorem.

### Weekly Subjects and Releated Preparation Studies

Week Subjects Preparation
1 What is a Curve? Arc-length, Reparametrization pp. 1-15
2 Level Curves vs. Parametrized Curves, Curvature Plane Curves pp. 16-34
3 Space Curves, Simple Closed Curve pp. 36-51
4 The Isoperimetric Inequality, The Four Vertex Theorem, What is a Surface? pp. 51-65
5 Smooth Surfaces, Tangents, Normals and Orientability, Examples of Surfaces pp. 66-82
6 Quadric Surfaces, Triply Orthogonal Systems, Applications of the Inverse Function Theorem pp. 84-95
7 Lengths of Curves on Surfaces, Isometries of Surfaces, Conformal Mappings of Surfaces pp. 97-111
8 Surface Area, Equiareal Maps and a Theorem of Archimedes, The Second Fundamental Form pp. 112-126
9 The Curvature of Curves on a Surface, The Normal and Principal Curvatures, Geometric Interpretation of Principal Curvatures pp.127-145
10 Gaussian and Mean Curvatures, The Pseudosphere, Flat Surfaces pp. 147-161
11 Surfaces of Constant Mean Curvature, Gaussian Curvature of Compact Surfaces, The Gauss Map pp. 161-169
12 Definition and Basic Properties of Geodesics, Geodesic Equations, Geodesic on Surfaces of Revolution, Geodesics as Shortest Paths pp. 171-196
13 Plateau’s Problem, Examples of Minimal Surfaces, Gauss Map of a Minimal Surface pp. 201-219
14 Gauss’s Remarkable Theorem, The Gauss-Theorem pp. 229-236, 247-267
15 Review
16 Final Exam

### Sources

Course Book 1. Elementary Differential Geometry, A. Pressley, Springer Verlag, 2000. 2. Differential Geometry of Curves and Surfaces, M. Do Carmo, Prentice-Hall, 1976. 3. Elements of Differential Geometry, R. S. Millman and G. D. Parker, Prentice-Hall, 1977.

### Evaluation System

Attendance/Participation - -
Laboratory - -
Application - -
Field Work - -
Special Course Internship - -
Quizzes/Studio Critics - -
Homework Assignments 4 10
Presentation - -
Project - -
Report - -
Seminar - -
Midterms Exams/Midterms Jury 2 50
Final Exam/Final Jury 1 40
Toplam 7 100
 Percentage of Semester Work 60 40 100

### Course Category

Core Courses X

### The Relation Between Course Learning Competencies and Program Qualifications

# Program Qualifications / Competencies Level of Contribution
1 2 3 4 5
1 Has the ability to apply scientific knowledge gained in the undergraduate education and to expand and extend knowledge in the same or in a different area X
2 Can apply gained knowledge and problem solving abilities in inter-disciplinary research X
3 Has the ability to work independently within research area, to state the problem, to develop solution techniques, to solve the problem, to evaluate the obtained results and to apply them when necessary X
4 Takes responsibility individually and as a team member to improve systematic approaches to produce solutions in unexpected complicated situations related to the area of study X
5 Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework X
6 Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility X
7 Has the ability to follow recent developments within the area of research, to support research with scientific arguments and data, to communicate the information on the area of expertise in a systematically by means of written report and oral/visual presentation X
8 To have an oral and written communication ability in at least one of the common foreign languages ("European Language Portfolio Global Scale", Level B2) X
9 Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge X
10 Follows scientific, cultural, and ethical criteria in collecting, interpreting and announcing data in the research area and has the ability to teach. X
11 Has professional ethical consciousness and responsibility which takes into account the universal and social dimensions in the process of data collection, interpretation, implementation and declaration of results in mathematics and its applications. X

Activities Number Duration (Hours) Total Workload
Course Hours (Including Exam Week: 16 x Total Hours)
Laboratory
Application
Special Course Internship
Field Work
Study Hours Out of Class 14 3 42
Presentation/Seminar Prepration
Project
Report
Homework Assignments 5 8 40
Quizzes/Studio Critics
Prepration of Midterm Exams/Midterm Jury 2 15 30
Prepration of Final Exams/Final Jury 1 20 20