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SF2722, Differential geometry, 7.5 hp, Spring 2009
Lecture
27/1: In
the first lecture of the course I tried to give some general
examples of ”local” and ”global” in analysis. We started
on Chapter 1 and the definition of a topological manifold. A lot
of the material in this chapter is not strictly needed for this
course, until next lecture you should read Chapter 1 with the
following priorities:
* Section
1.1-1.2: Important
until Exercise 1.2.11 (and the rest is also interesting.)
*
Section
1.3: Interesting,
but belongs rather to the course on topology.
* Section
1.4: The
main result (Corollary 1.4.12) is important, and we will return
to this later. You do not need to know the topological
techniques used in the proof.
* Section
1.5: Important,
we will return also to this material.
* Section
1.6: The
concept ”manifold with boundary” is a straight-forward
generalization of ”manifold”.
* Section
1.7: Fundamental
stuff, but belongs to the course on topology.
Lecture
3/2: We covered
Sections 2.1-2.3. Very important here are the abstract
characterizations of a tangent vector as ”equivalence class of
curves”(Def 2.2.4) or ”derivation on (germs of) functions at
point” (Def 2.2.15). The idea here is that the definitions of
a tangent vector at a point does not explicitly use the linear
structure of R^n, but only the knowledge which curves and
functions are smooth. You should read Section 2.2 very
carefully, leading to Cor 2.2.22. Note Lemma 2.2.20 which
contains an important factorization trick (which you might
recognize from algebra). Also read Section 2.3 carefully (a very
good exercise to check the understanding of 2.2!) Important are:
Thm 2.3.9 (the Chain Rule) and Exercise 2.3.10, definition of
”Diffeomorphism” (2.3.11), Thm 2.3.14.
Problem
session 10/2: We
solved some exercises on tangent vectors and differentials from
Section 2.3.
Lecture
10/2: In the third
lecture we discussed sections 2.4-2.7. In section 2.4 the
Inverse Function Theorem is stated (and the proof is given in
Appendix B). The philosophy is that a property of the
differential of a smooth map (which is a linear transformation)
is inherited by the map itself. A more general result of this
kind is the ”Constant Rank Thm” 2.4.6. The results here are
important in the next section where a submanifold of R^n is
defined. The sentence after Def 2.5.1 is important: not only
does a submanifold locally resemble R^r, it looks locally like
the standard embedding of R^r in R^n. This is perhaps more
information than we are interested in, and is one motivation for
the definition of a manifold which we will soon reach. As a test
of your understanding you should read Thm 2.5.3 and its proof
carefully. The tangent space of a submanifold of R^n is defined
in 2.5.6 (question: how would this be done in the ”derivation”
picture of tangent vectors?). In Section 2.6 smooth ”bump”
functions are constructed, and in Section 2.7 vector fields are
studied. Important here is the definition of ”Lie bracket”
(you must solve Exercise 2.7.9!) and the algebraic
characterization of vector fields as derivations on the algebra
of smooth functions (Thm 2.7.7 and all the steps of its proof.)
Problem
session 17/2: We
looked at two problems: showing explicitly that S^n is a
submanifold of R^(n+1) using stereographic projection, and we
made some computations of Lie brackets.
Lecture
17/2: In
section 2.8 vector fields are shown to generate a local flow.
The procedure of differentiating along the flow is called ”Lie
derivative” and is related to the Lie bracket by the important
Thm 2.8.16. Important is also Thm 2.8.19 which says that vector
fields commute if and only if their flows commute, you should
read the proof. Section 2.9 contains material slightly beyond
this course: Sard’s theorem (you need not read the proof but
you should read the formulation and examples on page 80), and
the Morse Lemma (which is of fundamental importance in
differential topology.) Make sure that you understand the
definition of the Hessian (Def 2.9.14) and why it makes sense
(you must solve Exercise 2.9.13!)
Problem
session 24/2: We
solved 2.8.13, and discussed maximal existence time for flows of
vector fields. Then 2.8.14 (a vector field is invariant under
its own flow), and 2.9.13, 2.9.16 (on the Hessian of functions).
Lecture
24/2: Now
we started on Chapter 3 with the definition of a smooth
manifold. Read very carefully section 3.1 where a smooth
structure on a topological manifold is defined as a maximal
atlas of coordinate charts. The two remarks on page 89 are
important: the transition maps between overlaps of different
charts constitute a cocycle, and it is possibly to construct M
given only this data. The definitions of smooth maps and tangent
spaces etc are nothing new, the constructions from Chapter 2
works just as weel on a smooth manifold (thanks to the smoothly
related charts) as on U in R^n. Section 3.2 contains some
comments on the possibility of a topological manifold to have
non-diffeomorphic smooth structures. Section 3.3 contains the
definition of the tangent bundle of a smooth manifold, this you
should also read carefully. The definition of a vector field is
very short (def 3.3.3), and it has the nice property that
smoothness is formulated wihtout referring to the component
functions of the vector field. The definition of a vector bundle
in general might seem a bit abstract (def 3.3.4), but we will
later see many examples of natural vector bundles over a
manifold. If you want you can skip section 3.4 completely, it is
an excursion into far more advanced material (we might return to
some of this at the end of the course when we discuss some
Riemannian geometry).
Problem
session 3/3: We
looked at an example of a smooth structure, the n-dimensional
torus T^n. Further we discussed covering spaces (Section 1.7,
Exercise 3.1.21) and Exercise 3.2.8 which gives an example of
distinct (=disjoint) smooth structures on the real line, which
nevertheless are diffeomorphic.
Lecture
3/3: We
continued the discussion of the Tangent bundle and checked in
detail how the bases and coordinates of tangent vectors change
under change of local chart of the manifold. This is not stated
in the course book, but can be found for example in the book
by Boothby, Corollary 1.8. In Section 3.5 the very important
”partitions of unity” are constructed. A partition of unity
can be used to take apart a globally defined object so that we
can compute with it as smooth pieces in local charts. It can
also sometimes be used to patch together local objects to a
globally defined object. We looked at the example of ”Riemannian
metrics” where this technique works (Exercise 3.5.9), and
”nowhere zero vector fields” where it does not work. Section
3.6 contains the definition of manifolds with boundary, which
you should read. We started (but did not finish) Section 3.7.
Here the important concepts are imbeddings and immersions. For a
short and clear treatment see Lecture
notes by Looijenga, p15-17.
Problem
session 10/3: We
discussed two things: First, the concept of ”manifold with
corners”, Exercise 3.6.6. This is inevitable if one wants to
work with smooth manifolds with boundary and take products of
manifolds. There is a procedure of ”straightening” or
”smoothing” the corner which gets rid of the corner set.
Second, we found an embedding of T² into R³ (an
extension of Problem 2 of homework 1.)
Lecture
10/3: Todays
sections were 3.7-3.9. In Section 3.7 we proved Theorem 3.7.11
which says that any compact smooth manifold of dimension n can
be embedded in R^k if k is chosen large enough. From this we
conclude that the abstract concept of ”smooth manifold”
introduced in Def 3.1.6 really has not given us any more objects
to play with than the ”submanifolds of R^k” from Def 2.5.1.
There is a stronger embedding result saying that k can always be
chosen as k=2n+1, this is the Whitney embedding theorem, Thm
3.7.12 (you should read the proof here!). In Section 3.8 the
very important concept of ”homotopy” is introduced, Def
3.8.1. Further there are results stating that smooth and
continuous homotopy are essentially the same, Thm 3.8.16 and Cor
3.8.18, in particular these apply to the definition of the
fundamental group (you need perhaps not be so careful with the
proofs in this section.) In section 3.9 we discussed the ”degree
mod 2” of smooth maps between manifolds of the same dimension,
up to the fundamental theorem of algebra in Thm 3.9.14.
Problem
session 17/3: First
we looked at Problem 1.5.5 concerning an embedding of the real
projective plane into R^4. Second we discussed the concept of
”winding number”, page 123, and how to extend this and the
degree to integer-valued invariants.
Lecture
17/3: From
Section 3.9 we saw how the Brouwer fixed point theorem (3.9.17)
follows from the simple observation that there can be no
retraction onto the boundary for a manifold with connected
boundary (3.9.16). Section 3.10 introduces the important concept
of ”Morse functions”; what they are (Def 3.10.2), they exist
(Thm 3.10.4), they are generic (Ex 3.10.18). Section 4.1 is on
flows of vector fields, the important result is that any vector
field on a compact manifold (or with compact support) is
complete, that is it has a flow defined for all values of
parameter time. Exercises 4.1.17-4.1.19 are important (question:
what are the limit sets of the gradient flow introduced in the
next section?) Section 4.2 contains many important ideas! First
we see how a Riemannian metric is used to associate a vector
field to the differential of a function (4.2.1). Then the flow
of this ”gradient field” is used to study the sub-level sets
M^a of the function f. When we slightly increase the value of
”a” two cases can occur: Either the sub-level sets are
diffeomorphic (Thm 4.2.3) if we do not pass a critical point of
the function, in this case the diffeomorphism can be constructed
from the gradient flow. Or the sub-level sets are related by
attaching a \lambda-handle (Thm 4.2.6), this happens if we pass
a critical point of index \lambda. You should read the proof of
4.2.6, and the important remark at the bottom of page 141. For
further information see Wikipedia
and the classic Morse
theory by J. Milnor.
Lecture
24/3 CANCELLED: Until
next week you can take a glance at the following sections which
we are interesting and important but not
included in the course:
* 4.5, Foliations and the Frobenius theorem, and
*
5, Lie groups and Lie algebras.
You should of course
also work on the homework
problems.
Lecture
31/3: Todays
sections were 6.1-6.3, introducing the cotangent bundle and
1-forms. In section 6.1 the concept of "dual bundle"
is introduced in a general formulation using cocycles. For a
more explicit treatment of change of coordinates etc for
the cotangent bundle see Boothby,
p 177-182 (in particular Cor. 1.7). Section 6.2 begins with
definition of T*M, important is the basis {dx^i}, and the
formula for the differential df in this basis. Read carefully
the discussion on "the Tensor Property" (Lemma 6.2.10,
Prop 6.2.11) on p188. This important principle states vaguely
that linearity over smooth functions is equivalent to being a
section of the cotangent bundle (or section of some other
appropriate bundle), one later application is to show that the
Riemannian curvature is a tensor (see Lemma 10.3.11). The two
main properties which makes T*M different from TM are the
existence of pullback by any smooth map (Lemma 6.2.12) and
exterior differentiation (Def 6.2.14). These operations commute
(Ex 6.2.16). In Section 6.3 the line integral of a 1-form along
a parametrized curve is introduced. The results of this section
should all be quite familiar. Exact forms are introduced in Def
6.3.4, and "locally exact" forms are introduced in Def
6.3.11. Later we will see that locally exact forms are the same
as "closed" forms. The discussion on page 194 (which
we did not have time for in the lecture...) is very important,
and you should read it carefully.
Tuesday
7/4: Easter,
no lecture.
Problem
session 14/4: Material
from 6.3, exact and locally exact 1-forms. We looked at the important
Example 6.3.12, and solved Exercise 6.3.14, showing that the integral
of a locally exact form around a closed loop only depends on the
homotopy class of the loop.
Lecture
14/4: Sections
6.4-6.5. The first (de Rham) comology space H^1(M) is introduced
in 6.4. We skipped the details of Prop 6.4.10, make sure to read
Lemmas 6.4.11-13 carefully. The important details in section
6.5 are the computations using 1-forms, such as the proof of Prop
6.5.1. But you should also read the second characterization of degree
(Prop 6.5.3) in terms of regular values. Lecture 21/4: Todays
subject was tensor fields and differential forms on a manifold, Chapter
7 in the book. Unfortunately the presentation in the course book is not
so easy to read, and a bit too advanced for this course. Instead I
recommend you to read Boothby, page 199-214. - Lecture 28/4: We discussed the exterior differential (d) and integration of differential forms. You should read Sections
8.1-8.2. Stokes's theorem is a generalization of the fundamental
theorem of calculus and links the operations d and integration. You
must read the proof of Thm 8.2.3, which is not complicated. Other
versions of Stokes's theorem are formulated with topological
applications in mind (Thm 8.2.9). In the lecture I discussed an
alternative characterization of orientation: an orientation on
n-dimensional M is given by a nowhere zero n-form. For this you should
check Boothby,
page 215-219. It is also a good idea to read Boothby page 219-223 for
exterior differentiation, page 236-240 for integration, and
251-266 for Stokes's theorem (note in particular Examples
5.2-5.4).
- Problem
session 5/5: Examples of computations with differential forms, integral, Stokes's theorem.
- Lecture 5/5: Today deRham
cohomology as a topological theory. The deRham cohomology vector spaces
are "topological" since they are isomorphic to the dual of singular
homology (with real coefficients), see Thm 8.2.21. The isomorphism is
given by the pairing defined by integrating a closed k-form over a
k-dimensional cycle (Prop 8.2.20). The Poincare Lemma tells us that
homotopic maps induce the same pullback maps on deRham cohomology (Thm
8.3.6), it gives the cohomology of contractible spaces (Thm 8.3.8,
8.3.9), and it tells us that the cohomology spaces measures the
difference between "locally exact" and "exact" (Thm 8.3.10). Pages
261-263 are about cohomology with compact supports which is parallel,
but slightly different (compare Thm 3.8.3 to Cor 8.3.17). Sections
8.4-8.9 further develop the topological theory. Highlights
are the Mayer-Vietoris sequence (Thm 8.5.2) which computes
cohomology of the whole in terms of cohomology of the pieces (Ex
8.5.10), and Cor 8.6.5, Thm 8.6.6, which relates orientability and
cohomology.
- Problem
session 12/5: We
discussed the Mayer-Vietoris sequence (Thm 8.5.2) for deRham cohomology
and used it to compute the cohomology spaces of spheres. Using similar
arguments it is not complicated to compute the cohomology of tori and
oriented surfaces (obtained by adding handles to S²).
- Lecture 12/5: Riemannian
geometry, Chapter 10. We are now beginning to describe how to add
geometric structure to a smooth manifold. In section 10.1 a connection
(or covariant derivative) on a manifold is introduced. With a
connection one can take the derivative of a vector field in the
direction of another vector field. The Levi-Civita connection of a
submanifold M in R^m (Def. 10.1.12) is an important example (note that
it coincides with the Levi-Civita connection of the induced metric on
M, Ex. 10.2.11). Given a connection on a manifold there is a covariant
derivative along curves (Thm 10.1.11), and the important concept of
parallel transport (Thm 10.1.13). In Section 10.2 Riemannian manifolds,
length of curves, and Riemannian volume forms are introduced. The
fundamental fact of Riemannian geometry is that a Riemannian
metric gives a unique torsion-free and metric connection. This
Levi-Civita connection is characterized by the Koszul formula (part (1)
of Ex. 10.2.12, you should derive this formula!). From the Koszul
formula follows the formula for the Christoffel symbols (or connection
coefficients) in local coordinates, see Wikipedia. (In Wikipedia we also find an interpretation of the torsion of a general connecion.)
- Problem
session 19/5: Computations with metrics and connections.
- Lecture 19/5: Riemannian
geometry 10.3-10.4. The Riemann curvature tensor is
introduced in Def. 10.3.14. The Riemann tensor is precisely the
obstruction to finding coordinates in which the riemannian metric has
constant coefficients (Thm 10.6.7). The information contained in the
full curvature tensor can be equivalently presented through the sectional curvature. By taking traces of the curvature tensor one obtains Ricci curvature and scalar curvature.
Ricci curvature is for example related to the behaviour of geodesics,
while scalar curvature is related to volume of small radius balls. The
curvature tensor is in general very complicated to compute, for a
submanifold of euclidean space the computations become quite
reasonable; The Gauss equation
expresses the curvature of the submanifold in terms of the Weingarten
map (or shape operator) and the curvature of the background space
(which vanishes for euclidean space).
Using the Gauss equations it is very easy to compute the curvature of
the standard sphere, the result is that the sphere has constant
positive curvature (and it is essentially the only space with this
property). In section 10.4 the concept of a geodesic in a Riemannian
manifold is introduced and the distance metric properties are
studied. In the Def 10.4.13 the distance function is defined, the
results 10.4.14-10.4.17 are very important. In particular the distance
between two points is always realized by a geodesic curve connecting
the points.
- Problem
session 26/5: Computations of curvature.
- Lecture 26/5: Riemannian curvature and geometry of manifolds.
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