Course Name Code Semester T+U Hours Credit ECTS
Manifoldlarin Diferensiyel Geometrisi MAT 007 0 3 + 0 3 6
Precondition Courses Students are assumed to be familiar with Differential Geometry I, II and Advances Differential Geometry.
Recommended Optional Courses
Course Language Turkish
Course Level Doctorate Degree
Course Type Compulsory
Course Coordinator Prof.Dr. MEHMET ALİ GÜNGÖR
Course Lecturers
Course Assistants
Course Category
Course Objective The Differential Geometry of Manifolds course aims to give the fundamental knowledge for the studies of graduate students who study at geometry branch.
Course Content Riemannian manifolds, Covariant differentiation, Curvature tensor, Theorem of Frobenius, induces connection and second fundamental form, Equation of Gauss, Codazzi and Ricci, Scalar curvature of submanifolds, minimal submanifolds in euclidean space, minimal submanifolds of a submaniforld, Examples of minimal submanifolds, Surfaces with paralel mean curvature, Surfaces with constant mean curvature in , Local existence theorem for surfaces with constant mean curvature, axiom of spheresLocus of spheres, Canal hypersurfaces, Ricci curvature and scalar curvature for pseudoumbilical submanifold, Characterizations of umbilical submanifolds, The Gauss map, geometric inequalities, total mean curvature, Submanifolds with nonnegative scalar curvature,
# Course Learning Outcomes Teaching Methods Assessment Methods
1 He/She synthesizes manifolds via differential geometry Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
2 He/She defines operation on manifold. Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
3 He/She computes the curvature of a manifold. Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
4 He/she illustrates minimal submanifold Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
5 He/She defines constant mean curvature in surface Lecture, Question-Answer, Group Study, Problem Solving, Testing, Homework,
6 He/She defines sphere axiom Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
7 He/She synthesizes surfaces via differential geometry Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
8 He/she illustrates hypersurfaces Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
9 He/She gives charecterization of a umbilicity submanifolds Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
10 He/she illustrates stable hypersurfaces Lecture, Question-Answer, Discussion, Group Study, Problem Solving, Testing, Homework,
Week Course Topics Preliminary Preparation
1 Riemannian manifolds, Curvature tensor, Theorem of Frobenius
2 Induced connection and second fundamental form
3 Equations of Gauss, Codazzi and Ricci
4 Scalar curvature of submanifolds
5 Minimal submanifolds in Euclidean space, Minimal submanifolds of a submanifold
6 Surfaces with parallel mean curvature vector
7 Surfaces with constant mean curvature in R^3 , Local existence theorem for surfaces with constant mean curvature
8 Axiom of spheres, locus of the spheres
9 Canal hypersurfaces
10 Ricci curvature and scalar curvature for pseudoumbilical submanifolds
11 The average fixed curvature pseudoumbilical submanifolds
12 Characterizations of umbilical submanifolds
13 The Gauss map, Geometric inequalities, Total mean curvature
14 Submanifolds with nonnegative scalar curvature
Resources
Course Notes 1. Şahin, B., Manifoldların Diferensiyel Geometrisi, Nobel Yayınları, Ekim 2012.
Course Resources 2. Chen, B., Geometry of Submanifolds, Marcel Dekker. Inc. New York, 1973.
3. Kobayashi, S., and Nomizu, K., Foundations of differential geometry, Number 15, Volume II, New York, 1969.
4. O’Neill B., Elementary Differential Geometry, Academic Press, New York, 1997.
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
1 At a master´s degree level, student reaches new knowledge via scientific researches, the use of knowledge of the same field as him/her or of different field from him/her, and the use of knowledge based on the competence in his/her field; s/he interprets the knowledge and prospects for the fields of application. X
2 Student completes the missing or limited knowledge by using the scientific methods. X
3 Student freely poses a problem of his/her field, develops a solution method, solves the problem, and evaluates the result. X
4 Student conveys, orally or in writing, his/her studies or the current developments in his/her field to the people in or out of his/her field. X
5 Student finds a solution to the unforeseen complex problems in his/her studies by developing new approaches. X
6 At a doctorate degree level, student prepares at least one scientific article of his/her field to be published in an international indexed journal, and s/he extends its popularity. X
7 Student analyzes the works that have been published before, approaches the same subjects with different proof methods, or determines the open problems about the current subject matters. X
8 Student looks for the scientists studying on the same field as him/her, and s/he gets in touch with them for a collaborative work. X
9 Student knows enough foreign language to do a collaborative work with the scientists studying on the same field as him/her abroad. X
10 Student follows the necessary technological developments in his/her field, and s/he uses them. X
11 Student looks out for the scientific and ethic values while gathering, interpreting and publishing data. 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
ECTS - Workload Activity Quantity Time (Hours) Total Workload (Hours)
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
Total Workload 146
Total Workload / 25 (Hours) 5.84
dersAKTSKredisi 6