Sergei Yakovenko's blog: on Math and Teaching

Friday, April 27, 2018

Lecture 5 (April 22, 2018)

Vector fields on manifolds and what they are good for

Various ways to formally introduce tangent vectors to an abstract object that apriori sits nowhere (via charts, derivations, etc.) How the tangent spaces get their structure of vector spaces. Flows and related stuff.

Commutator of vector fields. Mystery of the depressed order.

Flows and the Lie derivative. Lie bracket. Commutator revisited.

[Not covered in the class]: Commuting vector fields, commuting flows, common integral surfaces, involutive distributions, Frobenius theorem*.

The last couple of lectures deviated slightly from the lecture notes from the past years. For your convenience, here is a more digestible (?) version.


*The topic is briefly covered in the revised lecture notes. Please let me know (in comments), if you are interested in more explanations. Otherwise chances are that I will include the detailed proof (split into doable steps) as a problem for the exam (in which case you will learn the subject much better 馃槈 ).

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Lecture 3 (April 8, 2018)

Filed under: Calculus on manifolds course — Sergei Yakovenko @ 4:04
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Vector fields in open domains of \mathbb R^n

Ordinary differential equations, differential operators of the first order, local analysis and global consequences.

Sunday, March 25, 2018

Lecture 2. March 25, 2018

Differentiable maps

  • Definition of differentiability at a point. Maps f:U\to W between open subspaces of the Euclidean spaces U\subseteq \mathbb R^n,\ W\subseteq\mathbb R^m smooth on their domain.
  • Tangent spaces T_a U, tangent bundle TU=\bigcup_{a\in U}T_a U\simeq U\times\mathbb R^n.
  • Differential of a smooth map: \mathrm df:TU\to TW.
  • What is the derivative? (answer: exists only when n=m=1). Partial derivatives.
  • How do we define functions “having more than one derivative”?

Algebraic formalism:

  • Algebra C^\infty(U) of functions infinitely smooth in a domain U\subseteq\mathbb R^n
  • Pullback morphism of algebras f^*:C^\infty(W)\to C^\infty(U).

Vector fields: smooth maps v:U\to TU, such that v(a)\in T_a U.
Lie (directional, flow) derivations L_v:C^\infty(U)\to C^\infty(U). The Leibniz rule (algebra) and its meaning (“Any Leibniz linear map of C^\infty(U) to itself is a Lie derivative along some vector field).
Commutator of two vector fields (to be discussed more in the future).
Push-forward of vector fields by smooth invertible maps.

Monday, December 12, 2016

Lecture 6, December 12, 2016

Exterior derivation

The differential \mathrm df of a smooth function f is in a sense container which conceals all directional derivatives L_Xf=\left\langle\mathrm df,X\right\rangle along all directions, and dependence on X is linear.

If we consider the directional Lie derivative L_X\omega for a form \omega\in\Omega^k(M) of degree k\ge 1, then simple computations show that L_{fX}\omega is no longer equal to f\cdot L_X\omega. However, one can “correct” the Lie derivative in such a way that the result will depend on X linearly. For instance, if \omega\in\Omega^1(M) and X is a vector field, we can define the form \eta_X\in\Omega^1(M) by the identity \eta_X=L_X\omega-\mathrm d\left\langle\omega,X\right\rangle and show that the 2-form \eta(X,Y)=\left\langle\eta_X,Y\right\rangle is indeed bilinear antisymmetric.

The 2-form \eta is called the exterior derivative of \omega and denoted \mathrm d\omega\in\Omega^2(M). The correspondence \mathrm d\colon\Omega^1(M)\to\Omega^2(M) is an \mathbb R-linear operator which satisfies the Leibniz rule \mathrm d(f\omega)=f\,\mathrm d\omega+(\mathrm df)\land \omega and \mathrm d^2 f=0 for any function f\in\Omega^0(M).

It turns out that this exterior derivation can be extended to all k-forms preserving the above properties and is a nice (algebraically) derivation of the graded exterior algebra \Omega^\bullet(M)=\bigoplus_{k=0}^n\Omega^k(M).

The lecture notes are available here.

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. Kol谩r, P. Michor, J. Slov谩k, 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!
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* 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.

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