## Geometry Prelim topics

## Topics

Differentiable manifolds, smooth maps

Inverse function theorem, implicit function theorem, immersion, submersion

Partition of unity, embedding, Whitney embedding theorem

Sard’s 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.

## Main References:

- 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