Course Content (Syllabus)
Differentiable Manifolds (basic concepts). Fibre bundles. Covectors and 1-forms. Flow of a vector field and integral curves. Distributions. Frobenius theorem. Integral submanifolds. Basic concepts of foliations. Lie groups and Lie algebras (geometric consideration). Invariant vector fields. Integrations of Lie algebras and the exponential map. Examples.
Riemannian Metrics. Linear connections. Geodesics. Curvature. Riemannian submanifolds. Complete manifolds. Manifolds of constant curvature.
Additional bibliography for study
1) Loring W. Tu, An introduction to Manifolds, Universitext, Springer 2011.
2) John M. Lee, Introduction to Smooth Manifolds, GTM 218, Springer 2003.
3) D. Barden and Ch. Thomas, An Introduction to Differential Manifolds, Imperial College Press, 2003.
4) Lawrence Conlon, Differentiable Manifolds, Second Edition, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2008.
5) Frank W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Graduate Texts in Mathematics, 94, Springer-Verlag, New York-Berlin, 1983.