Course Information

Course Name	Turkish	Diferansiyel Geometri II
	English	Differential Geometry II
Course Code	MAT 610E	Credit	Lecture (hour/week)	Recitation (hour/week)	Laboratory (hour/week)
Semester	2	3	3	-	-
Course Language		English
Course Coordinator		Fatma Özdemir Fatma Özdemir
Course Objectives		1. To recall tensor fields, exterior derivative, differential forms and to introduce Lie derivative, connections, Riemannian metric and Riemannian manifold; 2. To examine covariant derivative, parallel translation, geodesics and normal coordinates and their properties; 3. To teach curvature tensors, sectional curvature, Ricci curvature and scalar curvature and to apply them to space forms; 4. To investigate conformal changes of Riemannian metric; 5. To introduce Riemannian submanifolds, induced connection and second fundamental form, and to obtain equations of Gauss, Codazzi and Ricci and Cartan structure equations.
Course Description		Tensor fields, exterior derivative, differential forms and Lie derivative. Connections. Riemannian metric, Riemannian manifold, covariant derivative, parallel translation, geodesics and normal coordinates. Curvature tensors, sectional curvature, Ricci curvature and scalar curvature. Space forms. Conformal changes of Riemannian metric. Riemannian submanifolds, induced connection, second fundamental form. Equations of Gauss, Codazzi and Ricci. Cartan structure equations.
Course Outcomes		Ph.D. students who successfully pass this course gain knowledge, skills and competency in the following subjects; I. Tensor fields, exterior derivative, differential forms and Lie derivative; II. Connections, Riemannian metric, Riemannian manifold; III. Covariant derivative, parallel translation, geodesics and normal coordinates; IV. Curvature tensors, sectional curvature, Ricci curvature and scalar curvature, space forms; V. Conformal changes of Riemannian metric; VI. Riemannian submanifolds, induced connection, second fundamental form; Equations of Gauss, Codazzi and Ricci; VII. Cartan structure equations.
Pre-requisite(s)		None
Required Facilities
Other
Textbook		1. Boothby, W.M. (1975). An Introduction to Differential Manifolds and Riemannian Geometry, Academic Press Inc.. 2. Hicks, N. J. (1971). Notes on Differentail Geometry, Van Nostrand Reinhold Company. 3. do Carmo, M.P. (1990). Riemannian Geometry, Birkehauser, 4. Kobayashi, S. ve Nomizu, K. (1963). Foundation of Differential Geometry I, Interscience Publishers. 5. Chen, B-Y,( 1973). Geometry of Submanifolds, M. Decker.
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