Geometry Prelim 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:

  1. Shigeyuki Morita, “Geometry of Differential Forms” Translations of Mathematical Monographs, Vol. 201, AMS
  2. John M. Lee, “Introduction to Smooth Manifolds” 2nd Edition, Graduate Texts in Mathematics, Springer.
  3. Frank Warner, “Foundations of Differentiable Manifolds and Lie Groups”, Graduate Texts in Mathematics, Springer.
  4. Michael Spivak, “A Comprehensive Introduction to Differential Geometry”, Vol.1, 3rd Edition.
  5. William M. Boothby, “An introduction to differentiable manifolds and Riemannian Geometry”, Academic Press.
  6. Ib H. Madsen and Jxrgen Tornehave, “ From Calculus to Cohomology”, Cambridge University Press.
  7. Loring Tu, An Introduction to Manifolds , Universitext