Pandit S.N. Shukla University, Shahdol (M.P) M.Sc. MATIHEMATICS SEMESTER-IV Paper V-Rlemannlan Geometry-II Syllabus

Pandit S.N. Shukla University, Shahdol (M.P) M.Sc. MATIHEMATICS SEMESTER-IV Paper V-Rlemannlan Geometry-II Syllabus

Theory Paper: Max. Marks- 42 

Internal Assessment: Max. Marks- 08 

Note: The question paper will consist of three sections A, B&C. Section A will consist of 7 objective ty pe questions each carrying I marks, section B will consist of 5 short answer type questions cach carry ing 3 marks and seetion C will consist of 2 long answer ty pe questions each carrying 10 marks. Section B& C will have internal choice. 

Unit 1- Delinition and examples of Differentiable manifolds. Tangent spaces. Jacobian mappings. 

Unit 2- Vector fields, Lie-bracket, Affine connections, Covariant derivatives, Curvature tensor, Bianchi identities. 

Unit 3- Lie-derivatives, Exterior algebra and Exterior derivative. 

Unit 4- Riemannian manifolds, Riemannlan connection, Curvature tensors, Sectional curvature, Schur's theorem. 

Unit 5- Geodesics in Riemannian manifolds, Projective curvature tensor, Conformal curvature tensor. 

Recommended Books: 

[1] R.S. Mishra, A course in Tensors with Applications to Riemannian Geometry, Pothishala Pvt. Ltd., Allahabad, 1965. 

[2] B.B.Sinha, Differential Geometry-An Introduction, Shyam Prakashan Mandir, Allahabad, 1978. 

Reference Books: 

[1] C.E.Weatherburn, An Introduction to Tensor Calculus and Riemannian Geometry, Cambridge University Press, London , 1942 and Radha Publishing House Calcutta, Indian Edition, 1995. 

[2] K.Yano, The Theory of Lie Derivatives and its Applications, North Holland Publishing Co. Amsterdam, 1957.

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