Sergei Yakovenko's blog: on Math and Teaching

Monday, January 30, 2017

Lecture 11 (Jan 16, 2017)

Filed under: Calculus on manifolds course,lecture — Sergei Yakovenko @ 3:54
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Geodesics

Definitions of geodesic curves on a Riemmanian manifold. Differential equations of the second order. Local existence of solutions. Geodesic map. Geodesic spheres, orthogonality. Local minimality of geodesic curves. Metric and geodesic completeness of Riemannian manifolds.

Survey of adjacent areas. Behavior of the nearby geodesics. Jacobi field. On surfaces: conjugated points, the role of the Gauss curvature, global properties of manifolds with positive and negative curvature (comparison). Hyperbolic plane \mathbb H, geodesics on it. Realization of the hyperbolic geometry on a surface in \mathbb R^3_{++-}. Impossibility of global embedding of \mathbb H into \mathbb R^3_{+++}.

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.

Monday, January 2, 2017

Lecture 8, Dec 26, 2016

De Rham and Cech cohomology of smooth manifolds

Using the exterior differential d on smooth differential forms and the fact that d^2=0, we define the de Rham cohomology with real coefficients H^k_{\mathrm dR}(M,\mathbb R) as the quotient space of closed k-forms by exact k-forms. This is a global invariant of a manifold M (for non-compact manifolds we may also consider a version for compactly supported forms, which yields different results).

De Rham cohomology can be computed using the Poincare lemma. If \mathfrak U=\{U_i\} is an open covering of M such that all opens sets and all their non-empty finite intersections are topologically trivial (homeomorphic to open balls), then for any closed form \omega\in\Omega^k(M) one can construct its primitives \xi_i\in\Omega^{k-1}(U_i) such that \mathrm d\xi_i=\omega in U_i. The (k-1)th forms \xi_i may disagree on the intersections U_{ij}=U_i\cap U_j, but one can attempt to twist them by suitable closed forms \mathrm d\phi_i. The corresponding system of (k-2)-forms \{\phi_i\} satisfies certain linear conditions on pairwise intersections; to satisfy these conditions one has to look for forms on triple intersections etc.

This construction gives rise to the notion of the Cech cohomology defined via systems of linear algebraic equations and reduces computation of the de Rham cohomology to a problem from linear algebra, determined by the combinatorics of the pairwise and multiple intersections of the sets U_i. In particular, one can conclude that the de Rham cohomology of compact manifolds is finite-dimensional.

An ultra-concise set of notes is available here (I hope to return and expand this text). The notion of Cech cohomology is further elaborated here.

NB. The class was shorter than usual because of the Hanukka lighting ceremony.

חנוכה שמח and Happy New Year, С наступающим Новым годом!

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