Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.
Definition and examples of manifolds; Submanifolds; Tangents; Coordinate vector fields; Tangent spaces; Dual spaces; Algebra of tensors; Vector fields; Tensor fields; Integral curves; Affine connections and Christoffel symbols; Covariant differentiation of tensor fields; Geodesics equations; Curve on manifold; Parallel transport; Lie transport; Lie derivatives and Lie Brackets; Geodesic deviation; Differential forms; Introduction to integration theory on manifolds; Riemannian Curvature tensor; Ricci tensor and Ricci scalar; Killing equations and Killing vector fields.
1. Bishop, R.L. and Goldberg, S.I., 1980.. Tensor Analysis on Manifolds. 1st ed. NY:
2. Carmo M.P., 1992. Riemannian Geometry. 1st ed. Boston:Birkhauser.
3. Lovelock, D. and Rund, H. Tensors., Differential Forms and Variational
4. Langwitz, D., Differential and Riemannian Geometry, Academic Press, 1970.
5. Abraham, R., Marsden, J.E. and Ratiu, T., Manifolds, Tensor Analysis and
Applications, Addison-Wesley, 1983.
6. Ali Shah. N, Vector and Tensor Analysis, 1992.
1. Ali Shah. N, Vector and Tensor Analysis, 1992.
2. Langwitz, D., Differential and Riemannian Geometry, Academic Press, 1970.
Sessional: 20 (Presentation / Assignment 10, Attendance 05, Quiz 05)
Mid-Term Exam: 30
Final-Term Exam: 50
Commencement of Classes October 12, 2020
Mid Term Examination December 14-12, 2020
Final Term Examination Febuary 02-08, 2021
Declaration of Result Febuary 19, 2021
Key Dates and Time of Class Meeting
Monday 3:30 pm-05:00 pm
Teusday 12:30 pm- 2:00 pm