Sergei Yakovenko's blog: on Math and Teaching

Monday, December 5, 2016

Lecture 5, Dec 5, 2016

Multilinear antisymmetric forms and differential forms on manifolds

We discussed the module of differential 1-forms dual to the module of smooth vector fields on a manifold. Differential 1-forms are generated by differentials of smooth functions and as such can be pulled back by smooth maps.

The “raison d’être” of differential 1-forms is to be integrated over smooth curves in the manifold, the result being dependent only on the orientation of the curve and not on its specific parametrization.

At the second hour we discussed the notion of forms of higher degree, which required to introduce the Grassman algebra on the dual space T^* to an abstract finite-dimensional linear space T\simeq\mathbb R^n. The Grassmann (exterior) algebra is a mathematical miracle that was discovered by a quest for unusual and unknown, with only slight “motivations” from outside.

The day ended up with the definition of the differential k-forms and their functoriality (i.e., in what direction and how they are carried by smooth maps between manifolds).

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. 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!
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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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