COURSE INFORMATION

Course Title Code Semester L+P

Hour Credits ECTS

RIEMANNIAN GEOMETRY MATH 512 1-2 3 + 0 3 10

Prerequisites

Language of

Instruction English

Course Level Graduate

Course Type

Course Coordinator Assoc. Prof. Dr. Ender Abadoğlu

Instructors

Assistants

Goals

To provide basic knowledge about Riemannian manifolds and curvature, to investigate the basic differential geometric structures related to the Riemannian metric.

Content

Riemannian manifolds. Absolute differentiation and connection. Riemann curvature, Bianchi identities. Geometry of hypersurfaces, Riemannian immersions and submersions. Completeness. Isometries and Killing vectors. Properties of curvature tensors.

Learning Outcomes Teaching

Methods

Assessment

Methods

1) Ability to make computations on Riemannian manifolds 1 A, B

2) Ability to analyze a geometric structures related to a

Riemannian manifold 1 A, B

3) Ability to make connections between curvature and

topology of a Riemannian manifold. 1 A, B

4) Ability to read a research article on the topic 1 B

Teaching

Methods: 1: Lecture, 2:Problem solving

Assessment

Methods: A: Written Examination, B: Homework

COURSE CONTENT

Week Topics Study

Materials

1 Review of Tensors, Manifolds and Vector Bundles Ch.2

2 Riemannian metrics and generalizations Ch.3

3 Connections Ch.4

4 Connections Ch.4

5 Riemannian geodesics Ch.5

6 Geodesics and distance Ch.6

7 Curvature Ch.7

8 Curvature Ch.7

9 Riemannian submanifolds Ch.8

10 Riemannian submanifolds Ch.8

11 The Gauss-Bonnet Theorem Ch.9

12 The Gauss-Bonnet Theorem Ch.9

13 Curvature and topology Ch.11

14 Curvature and topology Ch.11

RECOMMENDED SOURCES

Textbook Lee J.M., Riemannian Manifolds: An Introduction to Curvature, Springer,

1997..

Additional Resources Petersen P., Riemannian geometry, Second Edition, Springer, 2006.

MATERIAL SHARING

Documents

Assignments

Exams

ASSESSMENT

IN-TERM STUDIES NUMBER PERCENTAGE

Mid-terms

Quizzes

Assignments 5 100

Total 100

CONTRIBUTION OF FINAL EXAMINATION TO OVERALL

GRADE 50

CONTRIBUTION OF IN-TERM STUDIES TO OVERALL

GRADE 50

Total 100

COURSE CATEGORY

COURSE'S CONTRIBUTION TO PROGRAM

No Program Learning Outcomes Contribution

1 2 3 4 5

1 Acquires a rigorous background about the fundamental fields in mathematics and the topics that are going to be specialized.

x

2 Acquires the ability to relate, interpret, analyse and synthesize on

fundamental fields in mathematics and/or mathematics and other sciences. x

3 Follows contemporary scientific developments, analyses, synthesizes and evaluates novel ideas.

x

4 Uses the national and international academic sources, and computer and related IT.

x

5

Participates in workgroups and research groups, scientific meetings, contacts by oral and written communication at national and international levels.

x

6

Acquires the potential of creative and critical thinking, problem solving, research, to produce a novel and original work, self-development in areas of interest.

x

7

Acquires the consciousness of scientific ethics and responsibility. Takes responsibility about the solution of professional problems as a requirement

of the intellectual consciousness. x

ECTS ALLOCATED BASED ON STUDENT WORKLOAD BY THE COURSE DESCRIPTION

Activities Quantity Duration

(Hour)

Total

Workload

(Hour)

Course Duration (14x Total course hours) 14 3 42

Hours for off-the-classroom study (Pre-study, practice) 14 8 112

Mid-terms (Including self study)

Quizzes

Assignments 5 12 60

Final examination (Including self study) 1 36 36

Total Work Load

250

Total Work Load / 25 (h) 10

ECTS Credit of the Course 10