Geometry Prelim topics
Differentiable manifolds, smooth maps
Inverse function theorem, implicit function theorem, immersion, submersion
Partition of unity, embedding, Whitney embedding theorem
Smooth vector bundles, tangent vectors, vector fields and flows
Lie bracket, integrable distributions, Frobenius’ theorem
Basic Lie groups and Lie algebras
Cotangent bundle, differential forms, exterior differentiation, Lie derivatives, Cartan formula
Orientation, Integration on manifolds, Stoke’s theorem
De Rham cohomology, De Rham theorem
Riemannian metrics, geodesics, exponential map.
- Shigeyuki Morita, “Geometry of Differential Forms” Translations of Mathematical Monographs, Vol. 201, AMS
- John M. Lee, “Introduction to Smooth Manifolds” 2nd Edition, Graduate Texts in Mathematics, Springer.
- Frank Warner, “Foundations of Differentiable Manifolds and Lie Groups”, Graduate Texts in Mathematics, Springer.
- Michael Spivak, “A Comprehensive Introduction to Differential Geometry”, Vol.1, 3rd Edition.
- William M. Boothby, “An introduction to differentiable manifolds and Riemannian Geometry”, Academic Press.
- Ib H. Madsen and Jxrgen Tornehave, “ From Calculus to Cohomology”, Cambridge University Press.
- Loring Tu, An Introduction to Manifolds , Universitext