Syllabus

MA3H5 Manifolds

Autumn 2024

Week 1 (Sep 30-Oct 04)
Monday
  • Example: projective space.
  • Definition of pseudo-group of transformations.
Tuesday
  • Definition of a manifold.
Wednesday
  • Introduction to tensor products.
Week 2 (Oct 07-11)
Monday
  • Examples.
  • Open submanifolds, product manifolds, submanifolds.
Tuesday
  • More on submanifolds, manifolds defined by equations in another manifold.
Wednesday
  • Universal properties, tensor algebra.

Friday

(support class)
  • Definition of a manifold, smooth maps and diffeomorphisms.
  • Problems 1 and 4.
Week 3 (Oct 14-18)
Monday
  • Quotient manifolds.
Tuesday
  • Tangent vectors.
Wednesday
  • Symmetric algebra.
  • Introduction to exterior algebra.
  • Interpretation of exterior algebra via multilinear, alternating maps.

Friday

(support class)
  • Tensor products, local charts of manifolds.
Week 4 (Oct 21-25)
Monday
  • Tangent spaces, vector fields, bracket (beginning).
Tuesday
  • Differential, beginning of definition of immersion, submersion, embedding.
Wednesday
  • Immersions, submersions, embeddings.

Friday

(support class)
  • Review of tangent vectors and vector fields, commuting vector fields and P2-2.
Week 5 (Oct 28-Nov 01)
Monday
  • (Local) 1-parameter subgroups of (local) diffeomorphisms.
  • Flows.
Tuesday
  • Local flows and vector fields.
  • Complete vector fields.
Wednesday
  • Cotangent space (definition).
  • Smooth 1-forms, total differential, tangent bundle.

Friday

(support class)
  • Problem Sheet 2.
Week 6 (Nov 04-08)
Monday
  • Partitions of unity, part 1.
Tuesday
  • Partitions of unity, part 2.
  • Digression on analytic functions vs smooth functions.
Wednesday
  • Ehresmann's Theorem.

Friday

(support class)
  • Tangent vectors to surfaces in $\mathbb{R}^3$.
  • Riemannian metrics and pull-backs.
Week 7 (Nov 11-15)
Monday
  • Vector bundles.
  • Rank 1 bundles on $S^1$.
Tuesday
  • Differential $r$-forms.
  • Beginning of pull-backs of differential forms.
Wednesday
  • Pull-backs of differential forms.
  • Orientability.

Friday

(support class)
  • "Baby Pre-image theorem".
  • Tensor products and exterior squares.
Week 8 (Nov 18-22)
Monday
  • Integrals and Orientability.
Tuesday
  • Orientability of $S^n$, non-orientability of $\mathbb{P}^{2n}_{\mathbb{R}}$.
  • Exterior differentiation.
Wednesday
  • Exterior differentiation commutes with pull-back.

 

 

 

What we may be doing in the coming lectures: tentative syllabus

 

 

 

Last modified: Thursday, Nov 21 2024
This site is open source. Improve this page.