Sergei Yakovenko's blog: on Math and Teaching

Monday, February 6, 2017

Lecture 14 (Feb 6, 2017)

Calculus on complex manifolds

If V is a complex vector space, then it is naturally also a real vector space (if you allow multiplication by complex numbers, then that by real numbers is automatically allowed). However, forgetting how to multiply by the imaginary unit results in the fact that the dimension \dim_{\mathbb R}V of the space over the real numbers is two times higher. If we regret our decision to forget the complex multiplication, we still can restore it by introducing the \mathbb R-linear operator J\colon V\to V such that J^2=-E, where E is the identity operator.

An even-dimensional real vector space with such an operator is called an almost complex space, and it obviously can be made into a complex vector space (over \mathbb C). However, if we consider an even-dimensional manifold M with the family of operators as above, it is somewhat less than a complex analytic manifold (a topological space equipped with an atlas of charts with biholomophic transition functions). For details, follow the lecture notes that will be available later.

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Wednesday, January 11, 2017

Lecture 10, Jan 9, 2017

Introduction to the Riemannian geometry

  1. Flat structure of the Euclidean space and coordinate-wise derivation of vector fields.
  2. Axiomatic definition of the covariant derivative and its role in defining the parallel transport along curves on manifolds. Connexion.
  3. Covariant derivative \overline\nabla\text{ on }\mathbb R^n and its properties (symmetry, flatness, compatibility with the scalar product).
  4. Smooth submanifolds of \mathbb R^n. The induced  Riemannian metric and connection. Gauss equation.
  5. Weingarten operator on hypersurfaces and its properties. Gauss map.
  6. Curvatures of normal 2-sections (the inverse radius of the osculating circles). Principal, Gauss and mean curvatures.
  7. Curvature tensor: a miracle of a 2-nd order differential operator that turned out to be a tensor (“0-th order” differential operator).
  8. Symmetries of the curvature and Ricci tensors.
  9. Uniqueness of the symmetric connexion compatible with a Riemannian metric. Intrinsic nature of the Gauss curvature.

The lecture notes are available here.

Lecture 9, Jan 2, 2017

Sundries

I briefly discussed the (simplicial) homology construction in application to smooth manifolds and described several pairings: de Rham pairing (integration) between homology and cohomology, intersection form between H_k(M^n,\mathbb Z) and H_{n-k}(M^n,\mathbb Z), the pairing H^k_\text{dR}(M^n,\mathbb R)\times H^{n-k}_\text{dR}(M^n,\mathbb R)\to\mathbb R, \quad (\alpha,\beta)\longmapsto \displaystyle \int_M \alpha\land\beta and the Poincare duality.

Then I mentioned without proofs several results stressing the role of smoothness, in particular, how different smooth structures can live on homeomorphic manifolds. The tale of planar curve eversion and sphere eversion was narrated. For the video of the sphere eversion go here.

In the second part of the lecture I discussed natural additional structures that can live on smooth manifolds, among them

  • Complex structure, almost complex structure,
  • Symplectic structure,
  • Parallel transport,
  • Riemannian metric,
  • Group structure.

Then we prepared the ground for the next lecture, discussing how examples of these structures naturally appear (e.g., on submanifolds of the Euclidean space, on quotient spaces, …)

There will be no notes for this lecture, because of its mostly belletristic style.

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