Comments to lectures:
Subject: Differential geometry, lecture 16/1. Hi all, and thank you for attending the first lecture in the differential geometry course! The information on the course web page (http://www.math.kth.se/math/student/courses/5B1473/F/200607/) is now updated, in this mail I want to give some further comments. On the web page there are links to additional books and lecture notes: * Warner, Foundations of Differentiable Manifolds and Lie Groups, * Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, * Hitchin, Differentiable manifolds. In the first lecture we covered sections 1.1 and 1.2 (p. 1-9) in the course book containing the definitions of (topological and) differential manifolds and the definition of differentiable functions (and maps) on differential manifolds. The basic philosophy is to define a type of spaces where one has the possibility of interesting global phenomena, while the local structure is nice enough to allow techniques of analysis to be used. Make sure to understand Lemma 1.1.10 (on the existence of a maximal atlas) and Example 1.1.13 (differentiable structure on real projective spaces) properly! The example of projective space is interesting since it is a manifold truly defined through an atlas, and not via an embedding in Euclidean space (as with spheres). Next lecture we will see how submanifolds of R^n (curves and surfaces etc) are differential manifolds. Note that in this book the terminology "differential manifold" is used, where most authors use "differentable manifold". Alternative reading: Warner, p. 2-9; Boothby, p. 1-19 (introduction) p. 51-68 (manifolds); Hitchin, p. 2-12. Note that Hitchin has a slightly different definition where the topology is not assumed to exist but rather constructed from the differentiable structure (p. 7-8). (I mentioned the "Long line". One can read about this curiosity here: http://en.wikipedia.org/wiki/Long_line_(topology) This is an example of a space which is Hausdorff, locally euclidean, (and connected!) but not second countable.) Mattias
Subject: Differential geometry, lecture 23/1.
hi!
The second lecture was concerned with the concept of submanifold, sections
1.3, 1.4 of the book.
Important here is the rank of differentiable map (Def 1.2.7). Submanifolds
can easily be constructed from maps with "maximal rank", in two different
senses.
First in Def 1.3.6 we have the concepts of "regular value" and "submersion"
meaning maximal rank (=n) for f: M^m -> N^n, where m \geq n. In this case
the map is "locally surjective", and the inverse image of a regular value is
a submanifold.
Second in Def 1.3.11 we have the concept of "immersion" meaning maximal rank
(=m) when f: M^m -> N^n, m \leq n. Then the map is "locally injective", and
Thm 1.3.10 tells us that every point in M has a neighbourhood whose image is
a submanifold of N. However f(M) need not be a submanifold of N.
Lemma 1.3.12. is important since it gives conditions for when an immersion
is an embedding. Read the proof of this lemma! Note that it is generally
much simpler to check if a map is an immersion (via derivatives) than to
check that it is an embedding. Even simpler is using the observation at the
very end of section 1.3: An injective immersion of a compact manifold is an
embedding.
In section 1.4 (which I did not have time to mention much in the lecture)
it is shown that every ("abstract") manifold is diffeomorphic to a
submanifold of R^n for n large enough. Read the proof of Thm 1.4.3
carefully. This construction gives embeddings in R^n with n larger than
necessary. An exercise is to think about what n it gives for embedding the
sphere S^m or the torus T^m = (S^1)^m. What are minimal dimensions in these
cases?
Read the discussion after Thm 1.4.3.
Alternative reading:
Warner: p. 22-31 (slightly different definitions, prop 1.35 is our def),
Boothby: p. 69-81 (again slightly different definitions),
Hitchin: p. 9-11.
Mattias
Subject: Differential geometry, lecture 30/1. Hi! We are now in the middle of the (somewhat confusing) definitions of the tangent space of a smooth manifold. We will focus on two alternative definitions of tangent vectors: 1) as equivalence classes of smooth curves, and 2) as derivations acting on smooth functions. (There at least two other definitions/descriptions of interest.) In the lecture we covered the first half of Chapter 2, here some comments in detail: * Sections 2.1-2.2 give the definitions of tangent vectors as equivalence classes of smooth curves. Important is the identification (bottom of page 25) T_p M <--> R^n given a chart of M around p. Without this identification we do not know that tangent vectors form a vector space or that the derivative f_* is a linear map. When M = R^n and the chart is the identity map we get a (canonical) identification T_p R^n <--> R^n. As a further special case we get the differential df of a smooth function f:M->R, which is a cotangent vector. All these important concepts are on the middle of page 26, which you should read very carefully. * In Section 2.3 it is explained how the union of all tangent spaces T_p M forms a new manifold TM, the tangent bundle of M. Read pages 26-30. Important is the concept of "vector field" meaning a map associating a vector X(p) in T_p M to each p in M, in a way which is "smooth in p". One clever detail when viewing TM as a manifold on its own is that smoothness of X means just that X:M->TM is a smooth map, see def. 2.3.4. * Section 2.4 concerns the Whitney Embedding Theorem. This is important and you should at least read and think about the statement of the theorem. However, I do not consider this part of the course. * Section 2.5 contains the definition of tangent vectors as derivations, although not in much detail. Todays lecture finished somewhere around the middle of page 34, next time I will continue with more on vectors as derivations and how to compute things in bases for the tangent space. Different books on differential geometry give different "primary" definitions of tangent vectors. The derivation approach to tangent vectors is nicely explained in Boothby (p. 106-122) and Warner (p. 11-20). The first set of homework problems are now on the web page. Write your solutions in swedish or english. Mattias
Subject: Differential geometry, lecture 6/2.
hi!
With yesterdays lecture we are done with the definitions of tangent vectors
to a smooth manifold.
Viewing tangent vectors as derivations makes it convenient to write down
bases for the tangent and cotangent spaces at a point, given a chart around
that point. This is fundamental for computations but unfortunately not
explained in so much detail in the course book. I recommend you read:
* Warner p. 14-20:
- 1.19-1.24 (Definition of basis vectors for T_p M, T^*_p M, etc.)
* Boothby p. 106-120:
- 1.2 (Chain rule for f_*)
- 1.5 (Basis of T_p M, coefficients of vector X)
- 1.8 (Change of coordinates for tangent vectors)
- 1.9-1.10, 2.5 (good examples)
The first sentences on p. 112 in Boothby discuss tangent vectors defined
through their transformation properties. You should work out the change of
coordinates formula for bases {dx_i}, {dy_i} of the cotangent space, and for
the corresponding coefficients of a cotangent vector. (The result is that
while tangent vectors transform by multiplication with the Jacobi matrix of
the change of coordinates map, cotangent vectors change by multiplication
with the inverse matrix.)
In section 2.6 the Lie bracket is defined. You should verify that the
definition gives a vector field (or read Hitchins notes, p. 24-25). You
should also prove Lemma 2.6.2. and the local expression for the Lie bracket
on p. 38. Theorem 2.6.3. (Frobenius theorem) tells us that a set of vector
fields {X_i} are coordinate vector fields for some chart if and only if
[X_i,X_j] = 0.
Mattias
Subject: Differential geometry, lecture 13/2. Hi! The goal of yesterdays lecture was to go through section 2.7 in the book, and to understand Thm. 2.6.3, "Frobenius theorem". Since we ran out of time we will have to do the final steps next week. What we did is the following: * Integral curves of vector fields, local existence, global existence on compact manifolds, flow of vector field, R- (group-) action on a manifold. For this I recommend that you read Boothby, p.123-142, and Warner, p.36-40. * The Lie derivative of vector fields. Read Boothby, p.151-155, and Warner p.69-71 What is left is: * Show properly that two vectorfields have Lie bracket = 0 if and only if they have commuting flows. * Prove the Frobenius theorem. When we are done with this we will go on with chapter 4 in the book on differential forms and integration. The next set of homework problems will be handed out next week. Mattias
Subject: Differential geometry, lecture 27/2.
hi!
Today we did the last things from Chapter 2:
* Two vectorfields have Lie bracket = 0 if and only if they have commuting
flows. This can be found in Boothby, thm 7.12 p. 156 (the restriction to
small values of |t| and |s| is not necessary if the flows exist for all
time.)
* The Frobenius theorem as stated in the book. This is the Lemma on p. 161
of Boothby (with n=m).
Next we started on Chapter 4, but did not get further than the definition on
p. 66 of k-forms as alternating k-linear maps from a vector space to R
The reason why k-forms are natural to integrate over something k-dimensional
is that a change of parametrization automatically produces a determinant,
which is what is needed for change of variables in an integral.
We did an example with a 2-form, the computation to get the determinant goes
as follows. Suppose w is a two form on a vector space V, for X,Y in V we get
a number w(X,Y), so that w(X,X) = 0 or
w(X,Y) = -w(Y,X).
If we make a linear change of X,Y:
X' = aX + bY,
Y' = cX + dY.
then by multilinearity:
w(X',Y') = w( aX + bY, cX + dY )
= ac w(X,X) + ad w(X,Y) + bc w(Y,X) + bd w(Y,Y) .
Next since w is alternating we get
w(X',Y') = (ad - bc) w(X,Y),
what has appeared is the determinant of the linear change of vectors. If V
is the tangent space of a manifold and X,Y are tangent vectors to a
parametrized surface this shows that the integral of w over the surface does
not depend on the parametrization.
Next time we will continue with the full machinery of forms and integration
on a manifold.
Mattias
Subject: Differential geometry, lecture 6/3. Hi! Yesterdays lecture was on chapter 4, "Differential forms and integration". In section 4.1 the algebra of differential forms on a vector space is constructed. Prop 4.1.1 and prop 4.1.2 are both very important (as a little exercise you should write down the bases provided by 4.1.2 explicitly when V has dimension 2-4) The approach taken in this book is the shortest and most concrete: a differential form is just an alternating k-linear function on V = T_p M. A more sophisticated version (of exactly the same thing) can be found in Warner p.54-65, here differential forms are constructed together with tensors which are general multilinear functions on V and V^*. Also the discussion in Hitchin p.31-37 is very nice to read. (Note in particular the result on forms and determinant, prop 5.7, which gives a characterization of the determinant as the linear map induced on top degree forms, see also exercise 4.2 in the book) Section 4.2 defines forms on manifolds and the pull-back f^* by a smooth map f. The discussion (top of p.70) on forms expressed in a chart is important. In section 4.3 orientations of manifolds are defined. There are several equivalent characterizations of orientability, you should read prop 4.3.5 carefully. See also Warner p. 138-140 and Hitchin p. 53-55. Finally in section 4.4 the integral of an m-form over an oriented m-manifold is defined. The integral is defined by expressing the form in local coordinates and integrating. The above machinery comes together to show that this is independent of the chart. Remember to do the exercises on p. 76. Mattias
Subject: Differential geometry, lecture 13/3. Hi! In this weeks lecture we looked in detail on one example of integration over a manifold, the case of integration of the "standard" volume form integrated over the 2-sphere. We also defined the exterior derivative d, this is on p. 77-81 in the book. As usual we have a definition (5.2.1) in terms of a chart which is independent of the choice of chart (5.2.2). There is a second defintion (5.2.3) which has the advantage of not involving any chart, at the cost of using an expression which is not pointwise defined in the argument vectors. If one would use 5.2.3 as definition then it would be necessary to check that the expression only depends on the vectors X_i at a point p, even though the individual terms require the X_i to be vector fields around p. (Of course this follows when one checks that the definitions are equivalent.) In Hitchin, p. 41-42, the exterior derivatives of 1-forms and 2-forms on R^3 are computed, the result is related to the curl and div of vector fields on R^3. Next weeks lecture will be on Stokes theorem. Note that the lectures will be in seminar room 3733 at the math department from next week. Mattias
Subject: Differential geometry, lecture 20/3. hi! In this weeks lecture we continued with chapter 5. I mentioned that the exterior derivative is defined uniquely by four properties, but made a slight mistake when listing these properties. The fact that d commutes with pullback follows from the other properties. For the correct statement see Hitchins lecture notes, p. 41-45, or Warner, p. 65-69. After the definition of the exterior derivative d it is natural to consider forms in the kernel of d, "closed forms", and forms in the image of d, "exact forms". From the property d^2 = 0 it follows that every exact form is closed. The form (-y/r^2)dx + (x/r^2)dy on the plane without the origin is an example of a form which is closed but not exact. In section 5.3 manifolds with boundary are defined. In section 5.4 Stokes theorem is proved. From corollary 5.4.2 it follows that the above form is not exact. Mattias
Subject: Differential geometry, lecture 27/3. hi! this week we finished chapter 5, and started on chapter 6. Stokes theorem has a nice application in Prop. 5.4.3 which says that a compact oriented manifold cannot have a retraction onto its boundary. From this follows Brouwers fixed point theorem via a standard computation. We have already seen closed and exact forms. In chapter 6 deRham cohomology is introduced. This is a rule H^k which associates a vector space H^k(M) to every smooth manifold M and a linear map H^k(f): H^k(M) -> H^k(N) to each smooth map f: M -> N. There are different notations around for the linear maps H^k(f), (see p 99 in the book, p 49 in Hitchin, p 154 in Warner.) The very simple proposition 6.2.4 tells us that H^k is a functor. If f is a diffeomorphism it follows that H^k(f) is an isomorphism, so if H^k(M) and H^k(N) are not isomorphic then M and N cannot possibly be diffeomorphic (see Warner p 154). We looked at the simplest computations of deRham cohomology: * If M is connected then H^0(M) = R, and in general the dimension of H^0(M) is the number of components of M * H^1(S^1) = R can be computed explicitly using the fact that functions/ forms on S^1 correspond exactly to 1-periodic functions/forms on R. * H^n(M^n) = R if M^n is compact, connected, and orientable. The key point here is "orientable", using an orientation one defines a map H^n(M^n) -> R which is the integral over M of a form. This map turns out to be an isomorphism. The technical details are in section 5.5 which you should read. See also Hitchin, p 62-65. There will be no lecture on tuesday 3/4, next lecture is 10/4. Mattias
hi! In this weeks lecture we continued with deRham cohomology in sections in 6.2 and 6.3. In these sections the fundamental homotopy invariance of deRham cohomology is proved. If f,g:M -> N are homotopic maps then the induced maps on cohomology are equal, H^k(f) = H^k(g). The proof of this is divided in two parts. The algebraic part (6.2) is to define "cochain homotopies" and prove that cochain homotopic maps induce the same map on cohomology. The analytic part (6.3) consists of constructing a cochain homotopy between f^* and g^* when f,g:M -> N are homotopic. The main technical part here is Lemma 6.3.2 which you should read carefully since it is a good exercise in computations with forms. From homotopy invariance of cohomology one can draw strong conclusions, the first is that manifolds with different cohomology groups cannot be homotopy equivalent (we already knew that they are not diffeomoprhic). Examples 6.3.6 in the book are important, and Theorem 7.4 in Hitchin (p.58) is a very nice application. Mattias
hi! We have now finished chapter 6 on de Rham cohomology. In section 6.4 the Mayer-Vietoris sequence is derived. This is a tool to compute the cohomology of a manifold M in terms of the cohomology of two open sets covering M. For certain manifolds this makes it possible to compute the cohomology completely, in sections 6.5 and 6.6 the cohomology of spheres and tori is computed. Theorem 6.6.1 is a special case of the K�nneth formula which computes the cohomology of the product of two spaces. There are many different "cohomology theories" defined for different categories of spaces. The de Rham cohomology theory is isomorphic to any other cohomology theory (with real coefficients) which makes sense for smooth manifolds. The simplest formulation of such a result can be found in Warner, p. 154-155. The correct way to state that a functor is a cohomology theory is that it satisfies the Eilenberg-Steenrod axioms, and there is a general result saying that all such functors are equivalent. There is a short discussion of this in the book (p.150-151). One cohomological fact which is special to compact oriented manifolds is Poincare duality, for a statement of this see Warner p.226-227. There is a small mistake in the formulation of homework problem 4.1: one needs to assume also that N is connected. Mattias
Subject: Differential geometry, lecture 24/4. hi! In yesterdays lecture we looked at the degree of a smooth map between compact oriented manifolds. The degree picks out exactly one piece of data from the cohomology machinery, namely the induced map on top degree cohomology groups, which turns out to be an integer. The degree of a map can also be computed as a sum over the inverse image of a regular value, see thm. 7.1.2. We looked at a number of examples. The first were the same as exercises 7.1 and 7.2, p. 151. When we know that the map z -> z^n of the unit circle in C has degree n, and that the degree is invariant under homotopy of maps it is not hard to deduce that a complex polynomial without a root must be constant. Hitchins lecture notes have a nice discussion of degree, p. 62-69. Read also Thm 7.4 p. 58 (again) and formulate the proof as a statement on degrees of maps. In section 7.2 the linking number is defined as the degree of a certain map. For two linked circles in R^3 we find the Gauss integral for the linking number when we compute the degree as an explicit integral. This is Topological definition -> Analytic definition in: http://en.wikipedia.org/wiki/Gauss_linking_integral When we apply theorem 7.1.2 to compute the linking number we get a sum of +1/-1 over crossing points of a projection of the link, see: http://en.wikipedia.org/wiki/Linking_number Note that the procedure from thm 7.1.2 gives a sum over half the crossing points (only points where "K" is over "J"), in the Wikipedia reference it is summed over all crossing points and the result is divided by 2. Mattias
Subject: Differential geometry, lecture 15/5. hi! We are now going towards the end of the course. The remaining sections of the book (7.3-7.7) all deal with different ways of finding the Euler number (or Euler characteristic) of a manifold (even though it is not revealed until section 7.6 that we get the Euler number = alternating sum of Betti numbers.) In this weeks lecture we looked at the index of vector fields (7.3) and the Gauss map (7.4). I skipped over the details of the "Tubular neighbourhood thm", Lemma 7.4.1, but this is of course important. Most of the theory in these sections consists of playing around with the definitions. The technical step is concentrated to Thm 7.4.5 which is an application of Stokes Thm. The most important result is Thm 7.4.6, together with the corollary: I(X) does not depend on the vector field X, and deg(G) does not depend on the embedding of M in R^n. Next lecture we will continue with 7.5-7.6 together with some Riemannian geometry which fits nicely here. Mattias