## Differential Geometry(MAT417)

Course Code Course Name Semester Theory Practice Lab Credit ECTS
MAT417 Differential Geometry 8 4 0 0 4 8
 Language of Instruction French Course Type Compulsory Course Level Bachelor Degree Course Instructor(s) SUSUMU TANABE stanabe@gsu.edu.tr (Email) Assistant Objective The aim of the course is to provide the student with basic knowledge and skills in elementary differential geometry of curves and surfaces in local parametric treatment. Content Curves in R3: Frenet formulas and Fundamental Theorem. Regular surfaces. Inverse image of regular values. Differentiable functions on surfaces. Tangent plane; the differential of a map, vector fields, the first fundamental form. Gauss map, second fundamental form, normal, principal curvatures. Manifolds, tangent spaces and Lie bracket Course Learning Outcomes At the end of the course the student should be able to1. cope with modelling problems from diverse fields. 2. solve various problems in differential geometry and mechanics Teaching and Learning Methods Lectures, exercises References Millman, R.S. & Parker, G.D., Elements of Differential GeometryKühnel, W., Differential Geometry: Curves, Surfaces, ManifoldsEthan D. Bloch; A first course in Geometric Topology and Differential GeometrydoCarmo, M. Differential Geometry of Curves and SurfacesMontiel, S. & Ros, A. Curves and Surfaces
###### Theory Topics
Week Weekly Contents
1 Recall on smooth functions, inverse function theorem
2 Curves in the Euclidean space and their reparametrization
3 Tangent, normal and binormal vectors
4 Curvature and torsion of space curves
5 Fundamental theorem of curves
6 Surfaces in space and coordinate patches
7 Smooth surfaces
8 Tangent and normal vectors to a surface, first fundamental form and arc lengths
9 Second fundamental form and Weingarten endomorphisms
10 Normal curvature, mean curvature and Gaussian curvature
11 Theorema Egregium of Gauss and isometries
12 Gauss – Bonnet formula and its consequences
13 Manifolds and tangent spaces
14 Tangent spaces and Lie bracket
###### Practice Topics
Week Weekly Contents
Number Contribution
Contribution of in-term studies to overall grade 2 50
Contribution of final exam to overall grade 1 50
Toplam 3 100
###### In-Term Studies
Number Contribution
Assignments 7 0
Presentation 0 0
Midterm Examinations (including preparation) 2 50
Project 0 0
Laboratory 0 0
Other Applications 0 0
Quiz 0 0
Term Paper/ Project 0 0
Portfolio Study 0 0
Reports 0 0
Learning Diary 0 0
Thesis/ Project 0 0
Seminar 0 0
Other 0 0
Toplam 9 50
No Program Learning Outcomes Contribution
1 2 3 4 5
1 understands principles of deductive reasoning; has experience to verify well-foundedness and exactness of mathematical statements in systematic ways; X
2 can properly state and use concepts and results of major mathematical interest; X
3 masters current computational techniques and algorithms; has a good ability in their use; can identify relevant tools, among those one has learned, suitable to solve a problem and is able to judge whether or not one is in possession of these tools; X
4 is able to express one’s mathematical ideas in an organised way both in written and oral forms; X
5 understands relations connecting substantial concepts and results; can switch from one viewpoint to another on mathematical objects (pictures, formulae, precise statements, heuristic trials, list of examples,...); X
6 has followed individually a guided learning strategy; has pursued steps toward the resolution of unfamiliar problems; X
7 has a theoretical and practical knowledge in computer science well adapted for learning a programming language; X
8 has investigated the relevance of modeling and using mathematical tools in natural sciences and in the professional life; is conscious about historical development of mathematical notions; X
9 has followed introduction to some mathematical or non-mathematical disciplines after one’s proper choice; had experience to learn selected subjects according to one’s proper arrangement; X
10 masters French language as well as other foreign languages, to a level sufficient to study or work abroad. X
Class Hours 14 4 56
Working Hours out of Class 14 5 70
Assignments 7 3 21
Presentation 0 0 0
Midterm Examinations (including preparation) 2 6 12
Project 0 0 0
Laboratory 0 0 0
Other Applications 0 0 0
Final Examinations (including preparation) 1 16 16
Quiz 0 0 0
Term Paper/ Project 0 0 0
Portfolio Study 0 0 0
Reports 0 0 0
Learning Diary 0 0 0
Thesis/ Project 0 0 0
Seminar 0 0 0
Other 0 0 0