Course Name Code Semester T+U Hours Credit ECTS
Ileri Diferensiyel Geometri MAT 003 0 3 + 0 3 6
 Precondition Courses Students are assumed to be familiar with Differential Geometry I and II. Recommended Optional Courses Course Language Turkish Course Level yuksek_lisans Course Type Compulsory Course Coordinator Prof.Dr. MEHMET ALİ GÜNGÖR Course Lecturers Doç.Dr. MAHMUT AKYİĞİT, Course Assistants Course Category Course Objective The Advanced Differential Geometry course aims to give the fundamental knowledge for the studies of graduate students who study at geometry branch. Course Content Differentiable Manifolds, differentiable mappings, tangent vectors and tangent space, differentiation with respect to direction, parametric curve, Cotangent spaces, Covector, 1-form, duality, tangent vectors and tangent space on manifold, algebra of Class functions, Coordinate mappings on Tm(P), Riemannian metrics, Riemannian manifold, , differentiation with respect to direction and critic points, Hessian form of function, differentiability of a mapping, multi linear functions algebra, tensor algebra of spaces, tensors, covariant tensors, contravariant tensors, mixed tensors tensor algebra, symmetric tensors antisymmetric tensors, inner product of vector and tensor, symmetric product, symmetric algebra, Real exterior product space, properties of second order exterior product, example of a special exterior product, Isomorphic tensor spaces, tensor product of linear algebras and linear endorphism.
# Course Learning Outcomes Teaching Methods Assessment Methods
1 He/She defines the differentiable manifolds, Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
2 He/She adapts concepts of directional derivative and differentiation from analysis courses to directional derivative along a vector and differentiation on manifolds, Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
3 He/She adapts knowledge of algebra to functions of C^ -class, Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
4 He/She defines Riemannian metric and Riemannian manifolds, Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
5 He/She solves the problems related to Manifolds, Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
6 He/She defines tensors, Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
7 He/She classifies tensors, Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
8 He/She adapts knowledge of algebra to tensorlers and multi-linear functions, Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
9 He/She interprets the geometry with the aid of manifolds, Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
10 He/She relates mathematics and fundamental sciences to discipline of advanced differential geometry. Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
Week Course Topics Preliminary Preparation
1 Differentiable Manifolds, differentiable mappings, page 73-92
2 Tangent vectors and tangent space, differentiation with respect to direction page 92-112
3 Parametric curve, Cotangent spaces, Covector, 1-form, duality, tangent vectors and tangent space on manifold, page 112-127
4 Algebra of C^ -Class functions, Coordinate mappings on Tm(P), page 127-139
5 Riemannian metrics, Riemannian manifold page 139-143
6 Differentiation with respect to direction and critic points, Hessian form of function page 143-148
7 Differentiability of a mapping, page 148-162
8 Multi linear functions algebra, Tensor algebra of spaces, tensors page 162-174
9 Covariant tensors, contravariant tensors, mixed tensors, page 174-186
10 Tensor algebra, symmetric tensors, antisymmetric tensors page 190-201
11 Exterior product, exterior algebra, dimension of exterior algebra page 201-217
12 Inner product of vector and tensor, symmetric product, symmetric algebra, page 217-221
13 Real exterior product space, properties of second order exterior product, example of a special exterior product, page 221-232
14 Isomorphic tensor spaces, tensor product of linear algebras and linear endorphism. page 232-238
Resources
Course Notes [1] Hacısalihoğlu H. H. , Yüksek Diferensiyel Geometri, Fırat Üniversitesi, Fen Fakültesi Yayınları, Mat-No:2,1980.
Course Resources [2] Spivak Michael, Differential Geometry Volume1, Publish of Perisch, Houston, Texas,1999
[3] Spivak Michael, Differential Geometry Volume2, Publish of Perisch, Houston, Texas,1999
[4] Spivak Michael, Differential Geometry Volume3, Publish of Perisch, Houston, Texas,1999
Order Program Outcomes Level of Contribution
1 2 3 4 5
0 Develop strategic, political and practice plans and evaluate the results by taking into account the quality process in his/her area of expertise X
2 Student follows the current journals in his/her field and puts forward problems. X
3 Student understands the relations between the disciplines pertaining to the undergraduate programs of Mathematics X
4 Student gets new knowledge by relating the already acquired experience and knowledge with the subject-matters out of his/her field. X
5 Student uses different proof methods to come to a solution by analyzing the problems encountered. X
6 Student determines the problems to be solved within his/her field and if necessary takes the lead. X
7 Student conveys, in team work, his/her knowledge in the studies done in different disciplines by applying the dynamics pertaining to his/her own field. X
8 Student critically evaluates the knowledge got at the bachelor´s degree level, makes up the missing knowledge and focuses on the current subject-matters X
9 Student knows a foreign language to communicate orally and in writing and uses the foreign language in a way that he/she can have a command of the Maths terminology and can do a source research. X
10 Student improves himself/herself at a level of expertness in Mathematics or in the fields of application by improving the knowledge got at the bachelor´s degree level. X
11 Student considers the scientific and cultural ethical values in the phases of gathering and conveying data or writing articles. X
Evaluation System
Semester Studies Contribution Rate
1. Ara Sınav 70
1. Ödev 30
Total 100
1. Yıl İçinin Başarıya 50
1. Final 50
Total 100
Course Duration (Including the exam week: 16x Total course hours) 16 3 48
Hours for off-the-classroom study (Pre-study, practice) 16 3 48
Mid-terms 1 15 15
Assignment 1 10 10
Final examination 1 25 25