Sergei Yakovenko's blog: on Math and Teaching

Thursday, July 22, 2021


Filed under: Calculus on manifolds course,problems & exercises — Sergei Yakovenko @ 9:53

Welcome, try yourself!

The exam is posted below. It is pass/fail, no full solution of all problems is required, try to impress me by creative solutions and not to disappoint by grave mistakes. 
The results will be sent to FGS on August 22, 2021: don’t expect that I will be able to read your epistle within 48 hours, give me some breathing time to deal with your submissions.

Failures will not be reported. 
Scanned handwritten sheets are OK, but I would appreciate if you send me typeset solutions.

Good luck, and don’t hesitate to ask any questions, either here in comments or by email.
Exam problems.

Lecture 16, Jul 19, 2021

Filed under: Calculus on manifolds course — Sergei Yakovenko @ 9:28
Tags: , , ,

Sheaves, germs, Cech cohomology

In this concluding lecture we return to the basic notion of the “structural algebra” \mathscr C(M) of a manifold M which allows to introduce in the coordinate-free way all other geometric objects. Actually, the structural algebra is too large (and may be too rigid an object) to manipulate with. For instance, the property of a function to be invertible (a unit in the algebra) is rather rare in \mathscr C(M) (the function must be nonvanishing), but may well hold on very large parts of M where the function has no zeros.

Do deal with this, an idea of a sheaf was suggested. Morally sheafs are “functions with unspecified domains of definition”. Formally this means the following (we discuss only the sheaf of C^\infty-smooth functions).
We consider various coverings \{U_\alpha\} of the manifold M, and instead of a single algebra latex \mathscr C(M)$ we consider collection of algebras \mathscr C(U_\alpha), each of which should be thought of as the algebra of C^\infty-smooth functions defined in the open set U_\alpha. Clearly, these algebras cannot be totally independent. One obvious assumption is that if U_\beta\subseteq U_\alpha, then any element from \mathscr C(U_\alpha) can be “restricted” on U_\beta, i.e., there exists a natural morphism of algebras \mathscr C(U_\alpha)\to\mathscr C(U_\beta) with obvious composition rule for three morphisms associated with any triple U_\gamma\subseteq U_\beta\subseteq U_\alpha.
Given this “rectriction” morphisms, for any U_{\alpha,\beta} with a nonempty intersection U_{\alpha\beta}=U_\alpha\cap U_\beta, we may compare two elements f_\alpha\in\mathscr C(U_\alpha) and f_\beta\in\mathscr C(U_\beta) by looking at their restrictions on U_{\alpha\beta}. We say that these two elements agree on U_\alpha\cup U_\beta, if their restrictions coincide. Given a family of elements, we can generalize this notion for unions of several open sets.

Definition. Collection of algebras \mathscr C(U_\alpha) as above is called a sheaf, if any collection of elements \{f_\alpha\} which agree on the intersections, always corresponds to a unique element f\in \mathscr C(U), \ U=\bigcup_\alpha U_\alpha, which after restriction on each U_\alpha coincides with f_\alpha.

The flexibility contained in this construction allows to simplify formulations of many statements.

Definition. Let a\in M be a point on the manifold and \hat {\mathscr C}(a) a set of functions f, each defined (at least) in an open neighborhood U_f\subseteq M of a (each for each function) and C^\infty-smooth in this neighborhood. Two functions f,g\in \hat{\mathscr C}(a) are called equivalent, if there exists an open neighborhood U\subseteq U_f\cap U_g, \ U\owns a, such that f\equiv g on U. The equivalence classes are called germ of smooth functions, and the set of all germs is usually denoted by \mathscr C(M,a).

Theorem. Germs at a given point form a commutative \mathbb R-algebra with the natural operations. This algebra is local: the set \mathfrak m\in\mathscr C(M,a),\ \mathfrak m=\{f:\ f(a)=0\} is the unique maximal ideal in it.

The notion of a germ is the realization of the “maximal localization” of objects (functions, vector fields, differential forms) to a smallest, “indivisible” domain of definition which still follows to work with analytic tools (computing derivatives of any order, study convergence of the Taylor series etc.).

Scribbled notes:
Meeting Recording:

Access Passcode: 6$Fx25

Lecture 15, Jul 12, 2021

Complex manifolds

The field of complex numbers \mathbb C contains the real field \mathbb R as a subfield and geometrically is the real plane \mathbb R^2. This allows to do two operations: “forget” the complex structure and consider complex manifolds as real ones, and, on the other hand, “complexify” real manifolds to become complex (or rather to equip them with the complex structure).

Algebraically it is rather straightforward: for instance, one can take the \mathbb R-algebra \mathscr C(M) of C^\infty-smooth functions on a real manifold M and consider the tensor product \mathbb C\otimes_{\mathbb R}\mathscr C(M) which is a \mathbb C-algebra. Modifying this the principal algebra, we can proceed in this way to define other objects on the complex manifolds.

It turns out that the most convenient way to “remember” the complex structure on manifolds is to consider the involution “complex conjugacy” of \mathbb C which sends z to \bar z and fixes \mathbb R\subset\mathbb R. Let us start with the space \mathbb C^n and “forget” the complex structure: this would give us the space \mathbb R^{2n}. The operation of multiplication by \mathrm i=\sqrt{-1} is a linear operator J:\mathbb R^{2n}\to\mathbb R^{2n} such that J^2=-E which has eigenvalues \pm \mathrm i. To diagonalize J, we have to complexify we obtain \mathbb C^{2n} (this is what is always done in Linear Algebra when eigenvalues of the operator are non-real). As a result, we have a special basis in \mathbb C^{2n} with coordinates z_1,\dots,z_n,\bar z_1,\dots,\bar z_n and the involution “bar” on it.

Scribbled notes:
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Tuesday, July 6, 2021

Lecture 14, July 5, 2021

Symplectic geometry and Hamiltonian dynamics

In the same way as we introduced the Riemannian geometry as the geometry of manifolds additionally equipped with the symmetric positive definite bilinear form g:T_xM\to T_xM\to\mathbb R, we can (at the beginning only formally) look what happens if we replace in this construction the symmetric form by an exterior (antisymmetric) form \omega: T_xM\to T_xM\to\mathbb R, perhaps, with additional requirements. To what extent one extend this analogy?

As we know, any manifold can be equipped with a Riemannian metric. This is not the case anymore, but some manifolds come out naturally equipped with the antisymmetric form as above. It turns out that if N is a manifold of dimension n (a “configuration space” in the terminology of the classical mechanics), then its cotangent bundle M=T^*N=\bigcup_{x\in N}T^*_xN, a manifold of dimension 2n, admits a natural 2-form \omega\in \Omega^2(M) as above. Moreover, this form turns out to satisfy two extra assumptions:

  1. \omega is closed, \mathrm d\omega=0, and
  2. \omega is nondegenerate: if for some vector v\in T_aM, \ \omega(v,\cdot)\equiv 0 on T_aM, then v=0.

Definition. A manifold M equipped with a 2-form \omega\in \Omega^2(M) as above, is called a symplectic manifold.

The nondegeneracy condition immediately implies that a symplectic manifold must be even-dimensional.
Example. The Euclidean plane \mathbb R^2 with \omega=\mathrm dx\land\mathrm dy is a symplectic manifold.
This is the first obstruction for M to admit a symplectic structure.

Symplectic gradients

On Riemannian manifolds there is a natural isomorphism between tangent and cotangent spaces: any vector v is uniquely associated with a linear functional (covector) \xi which is the scalar product with v,\ \xi(w)=g(v,w). In such a way a function F:M\to \mathbb R with the differential \mathrm dF\in\Omega^1(M) gives rise to a vector field called the gradient of F, \mathrm{grad}F, such that g(\mathrm{grad}F, w)=\mathrm dF(w).

Replacing literally the Riemannian metric g by the symplectic form \omega, we arrive at the definition of a Hamiltonian vector field. If M is a symplectic manifold and H:M\to\mathbb R a smooth function, then the Hamiltonian vector vector field X_H is the unique (because of the non-degeneracy) vector field such that \omega(X_H,\cdot)=\mathrm dH\in\Omega^1(M).

Hamiltonian vector fields are in a special relationship with the differential form \omega and the mother function H: L_X\omega=0 for any H, and L_{X_H}H=0. Moreover, they form a Lie subgroup in the (infinite-dimensional) Lie group of vector fields \mathscr X(M) with respect to the commutator bracket. This makes the infinite-dimensional vector space \mathscr C(M) of functions on a symplectic manifold into a Lie algebra with Poisson brackets H,G\mapsto\{H,G\}.

The development of the story is in two independent directions. One can study dynamics of Hamiltonian vector fields, in particular, maximally symmetric vector fields (in the sense of the Lie algebra, possessing the maximal subalgebra of functions commuting with their Hamiltonians). This leads to a spectacular Liouville theorem and its deep generalizations known under the name of KAM (Kolmogorov-Arnold-Moser) theory.

The other direction is the study of diffeomorphisms between symplectic manifolds preserving the respective 2-forms. Darboux discovered that locally any two symplectic manifolds are equivalent to each other, in particular to a neighborhood of the Euclidean 2n-space with coordinates (x_1,\dots,x_n,p_1,\dots,p_n) and \omega=\sum_{i=1}^n \mathrm dp_i\land \mathrm dx_i. This is in sharp contrast with Riemannian manifolds which have an invariant (curvature tensor).

However, it turned out that despite this “local flexibility”, there are global invariants that constrain symplectomorhisms. One of the first was the celebrated Gromov’s non-squeezing theorem: if n\ge 2, than the unit ball cannot be squeezed inside a tube |x|\le \varepsilon in \mathbb R^4=(x_1,x_2,p_1,p_2) if \varepsilon is small enough.

Meeting Recording:
Access Passcode: 101n9U
Scribbled notes:

Wednesday, June 30, 2021

Lecture 13, June 28, 2021

Lie groups and Lie algebras

Lie groups are smooth manifolds with the additional structure of a group on them such that the group operations are smooth. There are not too many different (finite dimensional) Lie groups, but all of them are extremely important: most of them can be realized as subgroups of the groups GL(n,\mathbb R) or GL(n,\mathbb C) of square n\times n-matrices.

Lie groups are extremely homogeneous and rigid: for any two points a,b\in G there exists a unique (in fact, two) canonical map that diffeomorphically maps a neighborhood of a to a neighborhood of b, the left shift l_{ba^{-1}}:x\mapsto ba^{-1}x (the other map is the right shift r_{a^{-1}b}:x\mapsto xa^{-1}b). This allows to extend any geometric structure from an arbitrarily small neighborhood of any point, say, e\in G (the unity of the group), to global objects defined every where on G:

  1. Functions
  2. Vectors from T_e G=\mathfrak g
  3. Covectors from T_e^*G=\mathfrak g^*,
  4. Linear self-maps \mathfrak g\times\mathfrak g\to\mathfrak g

and many other structures. As a result, we create left-invariant global objects (functions, vector fields, 1-forms from \Omega^1(G) e.a.). Of course, left-invariant functions are only constants, but other objects may be quite nontrivial. In particular, we obtain for granted a connexion (“parallel transport”) between any two tangent spaces, which is flat (independent of any path between two respective points). Its curvature is zero, but its torsion is not: [\nabla_X,\nabla_Y]=-[X,Y].

What is important is that the correspondence

left-invariant objects on G\Longleftrightarrow objects on the tangent space \mathfrak g=T_eG

allows to translate analytic constructions involving calculus (derivations, external derivatives, commutator of vector fields) to purely algebraic constructions on \mathfrak g.

The principal construction is the commutator of vector fields on \mathscr X(G), which equips \mathfrak g with an algebraic operation called the Lie bracket \mathfrak g\times\mathfrak g\to\mathfrak g,\ v,w\mapsto[v,w]. This operations is bilinear, antisymmetric and satisfies the Jacobi identity [u,[v,w]]+[v,[w,u]]+[w,[u,v]]=0. A linear space equipped with such bracket is called a Lie algebra (note that this algebra is non-associative, contrary to the usual assumption on abstract algebras).

The opposite construction which associates with an abstract Lie algebra a Lie group is called the exponentiation. It is not always well defined, but morally the Lie bracket operation “remembers” all what is necessary to reconstruct the Lie group from its tangent space \mathfrak g.

Yet it should be noted that finite-dimensional Lie groups and algebras are like precious gems: they are rare and shine everywhere, but the real building blocks are infinite-dimensional analogs. For instance, all diffeomorphisms of an (abstract) smooth manifold M form an infinite-dimensional Lie group, and the space of smooth vector fields \mathscr X(M) is an infinite-dimensional Lie algebra with the bracket operation being the same old commutator.

Scribbled notes

Video record:

Access Passcode: CkfUC9

Saturday, June 26, 2021

Lecture 12, June 21, 2021

Geodesic lines (curves)

Geodesics can be defined in two different ways. On a manifold with an abstract connexion (covariant derivative) they are self-parallel curves: \nabla_v v=0,\ v=\dot\gamma(t), the velocity vector is parallel along \gamma. This definition corresponds to the equation of a free particle \ddot x(t)=0 moving in absence of external forces.

The second definition is related to the problem of length minimization: geodesic curve realizes the minimal length between any two its points sufficiently close to each other.

We derived second order differential equations for the geodesics and studied behavior of their solutions depending on the Riemann curvature (in the 2D case).

Scribbled notes:
Meeting Recording:
Access Passcode: 3&eDHS

Tuesday, June 15, 2021

Lecture 11, June 14, 2021

Covariant derivative. Gauss equations for hypersurfaces

We again discussed the relationship between (abstract) parallel transport and covariant derivatives, differential operators of certain form, generalizing the Lie (directional) derivative of scalar functions.

We computed the Riemannian connexion for hypersurfaces of the Euclidean space \mathbb R^{n+1} and introduced (very briefly) the curvature tensor of the corresponding induced Riemannian metric. Its full role will become more clear when we discuss behavior of the geodesic curves on Riemannian surfaces.

Meeting Recording:
Access Passcode: 63yWpn
Scribbled notes:
(I corrected the error in the printed notes spotted by Jonathan).

Wednesday, June 9, 2021

Lecture 10, Jun 7, 2021

Riemannian manifolds

Definition, examples, quick facts. Behavior of Riemannian metric by maps, isometries, embeddings. Lengths, angles. Geodesic curves. 

Flat manifolds, parallel transport. Parallel transport along large circles on the round sphere \mathbb S^2\subseteq\mathbb R^3. Curvature (first taste).

Hypersurfaces of the Euclidean space. Infinitesimal parallel transport along curves.

Meeting Recording:
Access Passcode: d$L6Y7

Scribbled notes:
Draft of a textbook for high school teachers (Rothschild-Weizmann program):

Monday, May 31, 2021

Lecture 9, May 31, 2021

Stokes theorem. Cohomology and homology

Having introduced the operators \mathrm d,\partial of the exterior differential on forms and the boundary on the smooth chains, we prove that

\displaystyle \int_{\partial \sigma}\omega=\int_{\sigma}\mathrm d\omega

for all chains and forms of appropriate rank/dimension.

The identities \mathrm d^2=0,\ \partial^2=0 imply that each exact form \omega=\mathrm d\eta is closed, \mathrm d\omega=0 (and respectively for chains without boundaries/chains that are themselves a boundary). How large are the quotients?
We introduce the (de Rham) cohomology and (simplicial) homology and look at the first examples of their computation.

Meeting Recording:
Access Passcode: KA+Kd8
Scribbled boards (for the last two lectures):

Sunday, May 30, 2021

Lecture 8, May 24, 2021

Integration of exterior forms. Chains, orientation and boundaries

The lecture was devoted to purely topological questions, how to accurately define a multidimensional generalization of a compact closed line segment for higher dimension k so that k-forms could be integrated over such objects in the same way as 1-forms can be integrated along oriented smooth paths.

Meeting Recording:

Access Passcode: eM3UU%

The scribbled boards will be posted after the next lecture as a single file for the two lectures.

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