Sergei Yakovenko's blog: on Math and Teaching

Wednesday, November 30, 2016

Lecture 4, Nov 28, 2016

Objects that live on manifolds: functions, curves, vector fields

We discussed how one may possibly define smooth functions on manifolds, smooth curves, tangent vectors, smooth vector fields. Next we discussed how these objects can be carried between manifolds if there exists a smooth map (or diffeomorphism) between these manifolds.

Flow of vector field. Lie derivatives.

Every vector field X on a compact smooth manifold M defines a family of automorphisms F^t_X (diffeomorphic self-maps) of M which form a one-parametric group, called the flow. Any object living on M can be carried by the flow by the operators \bigl(F^t_X\bigr)^*, t\in\mathbb R. The Lie derivative along X is the velocity of this action at t=0, namely, L_X=\frac{\mathrm d}{\mathrm dt}\big|_{t=0}\bigl(F^t_X\bigr)^*.

We show that the Lie derivative of functions coincides with the action of the corresponding derivations, and the Lie derivation of another vector field is the Lie bracket L_XY=[X,Y].

At the end of the day we establish the identities [L_X,L_Y]=L_{[X,Y]} and the Leibniz rule for L_X with respect to the Lie bracket, L_X[Y,Z]=[Y,L_XZ]+[L_XY,Z]. Both turn out to be equivalent to the Jacobi identity [X,[Y,Z]]+[Y,[X,Z]]+[Z,[X,Y]]=0 for the Lie bracket.

The lecture notes are available here.

Further reading

In addition to previously mentioned books, you may like the book I. Kolár, P. Michor, J. Slovák, Natural Operations in Differential Geometry, freely available from the Web.

Besides, I mentioned that the Jacobi identity has many different faces. One of them, discovered by V. Arnold, can be stated as follows: the three altitudes of a triangle intersect at one point because of the Jacobi identity*. You can find the explanations here and here. Enjoy!
________________________________________
* In fact, it is a slightly different Jacobi identity, not for the Lie bracket of vector fields, but for the vector product \mathbb R^3\times\mathbb R^3\mapsto\mathbb R^3, u,v\mapsto [u,v]=u\times v. But later we will see that this vector product is the commutator in the Lie algebra of vector fields on the group of orthogonal transformations of \mathbb R^3, thus the difference is purely technical.

Advertisements

Monday, November 21, 2016

Lecture 3, Nov 21, 2016

Filed under: Calculus on manifolds course,links — Sergei Yakovenko @ 4:51
Tags: ,

Concept of Manifold

The entire lecture was devoted to motivation and examples of C^\infty-smooth manifolds (submanifolds of \mathbb R^n, spheres, tori, projective spaces, matrix groups etc).

Slightly more detailed plan of the lecture is here.

If you want to read more (which is most highly welcome), here are a few recommendations:

  1. F. Warner, Foundations of Differentiable Manifolds and Lie Groups
  2. W. M. Boothby, Introduction to Differentiable Manifolds and Riemannian Geometry
  3. N. J. Hicks, Notes on Differential Geometry, Van Nostrand
  4. B. A. Dubrovin, Differential Geometry. Notes from SISSA course (Trieste, Italy)

A few thoughts on how to use these books. The subject (calculus on manifolds) is difficult because it involves both complicated concepts and the new language describing these concepts, and there is no way to learn these things but in parallel. One possibility to practice in the new language is to read as many texts about familiar subjects, as possible. This is what I suggest: if you believe you understand certain things, try to read about them in different books and make sure that different notation adopted by different authors does not detract you from the core.

A little bit more specific note. A closely related beautiful subject, Algebraic Geometry, was born from studies of how subsets of real or complex Euclidean space may look like. For some time it developed using mostly geometric/analytic tools, but eventually it was realized that to avoid problems with singularities, “double points”, “points at infinity” etc., one should start with the algebra of polynomials in one and several variables, its ideals, the quotient algebras and schemes in general. This approach brought tremendous achievements.

In my attempt to present the basic constructions of Calculus on Manifolds and, more generally, Differential Geometry, I decided to make the first several steps in a similar spirit and build objects from the algebra of C^\infty-smooth functions on a manifold. Of course, these algebras are very different from the algebras of polynomials (in particular, they are not Noetherian), which makes life some times easier, some times more difficult.

See you in a week.

Tuesday, November 15, 2016

Lecture 2 (Nov. 14, 2016).

Filed under: Calculus on manifolds course — Sergei Yakovenko @ 5:07
Tags: , ,

Tangent vectors, vector fields, integration and derivations

Continued discussion of calculus in domains lf \mathbb R^n.

  • Tangent vector: vector attached to a point, formally a pair (a,v):\ a\in U\subseteq\mathbb R^n, \ v\in\mathbb R^n. Tangent space T_a U=\ \{a\}\times\mathbb R^n.
  • Differential of a smooth map F: U\to V at a point a\in U: the linear map from T_a U to T_b V,\ b=F(a).
  • Vector field: a smooth map v(\cdot): a\mapsto v(a)\in T_a U.  Vector fields as a module \mathscr X(U) over C^\infty(U).
  • Special features of \mathbb R^1\simeq\mathbb R_{\text{field}}. Special role of functions as maps f:\ U\to \mathbb R_{\text{field}} and curves as maps \gamma: \mathbb R_{\text{field}}\to U.
  • Integral curves and derivations.
  • Algebra of smooth functions C^\infty(U). Contravariant functor F \mapsto F^* which associates with each smooth map F:U\to V a homomorphism of algebras F^*:C^\infty(V)\to C^\infty(V). Composition of maps vs. composition of morphisms.
  • Derivation: a \mathbb R-linear map L:C^\infty(U)\to C^\infty(U) which satisfies the Leibniz rule L(fg)=f\cdot Lg+g\cdot Lf.
  • Vector fields as derivations, v\simeq L_v. Action of diffeomorphisms on vector fields (push-forward F_*).
  • Flow map of a vector field: a smooth map F: \mathbb R\times U\to U (caveat: may be undefined for some combinations unless certain precautions are met) such that each curve
    \gamma_a=F|_{\mathbb R\times \{a\}} is an integral curve of v at each point a. The “deterministic law” F^t\circ F^s=F^{t+s}\ \forall t,s\in\mathbb R.
  •  One-parametric (commutative) group of self-homomorphisms A^t=(F^t)^*: C^\infty(U)\to C^\infty(U). Consistency: L=\left.\frac{\mathrm d}{\mathrm dt}\right|_{t=0}A^t=\lim_{t\to 0}\frac{A^t-\mathrm{id}}t is a derivation (satisfies the Leibniz rule). If A^t=(F^t)^* is associated with the flow map of a vector field v, then L=L_v.

Update The corrected and amended notes for the first two lectures can be found here. This file replaces the previous version.

Monday, November 7, 2016

Lecture 1 (Nov 7, 2016)

Crash course on linear algebra and multivariate calculus

Real numbers as complete ordered field. Finite dimensional linear spaces over \mathbb R. Linear maps. Linear functionals, the dual space. Linear operators (self-maps of linear space), invertibility via determinant. Affine maps, affine spaces.

Polynomial nonlinear maps and functions, re-expansion as a tool to construct linear (affine) approximation. Differential. Differentiability of maps, smoothness of functions.

Inverse function theorem.

Vector fields, parameterized curves, differential equations.

The first set of notes is available here here.

Calculus on manifolds: new course announcement

Filed under: Calculus on manifolds course — Sergei Yakovenko @ 4:19

‘שלום כיתה א

This academic year (תשע”ז) after a long pause I will “again” teach “this” course (which previously was delivered under the name of “Differential Geometry”). This time I decided to give it the name more appropriate for the content.

Each week some 90+ minutes of lecture will take place in Room 155, Zyskind Building every Monday 10:15–12:00, starting from November 7, 2016 and the last lecture scheduled for February 6, 2017. Then there will be a take-home exam with about a month to submit solutions.

I will post the lecture notes on this blog: to simplify search, look at the respective category. Usually notes will be prepared before the talk, but to keep records straight I will also briefly summarize which parts I succeeded to cover in the real 90+ minutes of time.

Previous courses’ notes, some reading recommendations and other relevant information are available from the stationary repository.

You are most welcome to ask questions, answer them, report grave errors in the notes and interact with this blog in any other way. Please, no likes! 😉

Create a free website or blog at WordPress.com.