TMA4190 Manifolds spring 2011

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 18 May I hope you have studied well on your own during all the passed weeks.You need to understand the concepts and play with them by studying various exercises !!! Tuesday April 12 and Thursday April 15 are the last time of practice session and lecture, respectively. We shall look at various exercises, including previous exam problems, as well as discussion of various concepts related to these. A note on basic Riemannian geometry is available under Pensum. Also some basic ideas of differential topology is included for convenience. A list of the basic concepts is presented, as an overview. See Pensum.

Read about the Lie bracket of vector fields, see p.37-38 in Barden-Thomas. We shall discuss its meaning, some of its properties, and do some calculations. See also the exercises for Tuesday March 29 session. We shall also discuss what is the difference between an orientable , oriented, or not-orientable manifold. But after this, we shall only calculate and repeat the things covered before, looking at various exercises. The students are also invited to propose topics we should repeat in the remaining lectures|

19 March

We are presently working with the following concepts 1) dynamical system on a manifold M, 2) vector field on M,3) (global or local) flow on M, 4) (global or local) one-parameter group on M. We give examples, and will demonstrate the intimate connection between these four concepts. We have already discussed Riemannian manifolds, that is, manifolds with a Riemannian metric. We have also seen that a submanifold of a Riemannian manifold inherits an induced Riemannian metric and thus becomes a Riemannian manifold. A good example is the round 2-sphere in the Euclidean 3-space. The isometry group of a Riemannian manifolds may be regarded to be its "symmetry" group. After all this, we have gone through essentially the whole curriculum of the course, which we shall specify more in detail later.The lectures last until Thursday 14 April, and we shall use much of the remaining time to work with exercises, which also serve as a good repetition.

 11 March We have discussed what is a Riemannian metric on a manifold. The textbook is not so instructive about this topic, whereas the approach in Dundas is to define the metric as some cross section (with certain properties) of a certain vector bundle. But we shall be more elementary and look at various examples. You can find many kinds of texts via Google about Riemannian metrics. Just try, for example http://en.wikipedia.org/wiki/Riemannian_manifold#Examples

So finding texts about this topic is quite simple, without using any book. We do not have much more new material left. It remains to discuss the connection between the topics: 1) a vector field, 2) a one-parameter group (or flow), 3) a dynamical system (1.order ODE).

 4 March We have defined the tangent bundle TM of a smooth manifold M, and how a smmoth map between M and N induces a smooth map (called the derivative) between TM and TN. TM is a smooth manifold of dim 2m, and it is also an example of a vector bundle. We shall continue going through the basic theory of vector bundles, but the main focus will be the idea behind the tangent bundle.

From now on we shall continue discussing vector fields and covector fields (also called 1-forms) on M, and we shall also see how a vector field gives a dynamical system (1.order ODE) on M, and conversely how a dynamical system yields a vector field. In fact, we shall see that the following four concepts : (1) a vector field on M, (2) a dynamical system on M, (3) a 1-parameter group on M, and (4) a flow on M, are (essentially) equivalent concepts, in the sense that the same basic idea manifests itself in four different ways. All this is discussed in Chap.2 of [Barden–],but rather briefly (as usual). You will also find all this stuff in [Dundas], but you will have to search in the appropriate chapters yourself.

 24 February Section 1.3 in [Barden&Thomas] is probably difficult to read, especially the formulation and use of the implicit function theorem. The topic in this section is more readable in [Dundas], where the implicit function theorem is not used. The lectures have followed [Dundas] on these topics. Today we shall start more systematically with the "derivative" of smooth maps, which is the beginning of chap. 2 in [Barden&Thomas], or chap. 4 in [Dundas]. The tangent bundle (and vector bundles in general) is covered by Dundas chap.5, and appears in section 2.3 and 3.2 of [Barden&Thomas]. But the latter book discusses fibre bundles more generally, and thus vector bundles as a special case, but in fact it is just as easy (?) to learn the definition of a fibre bundle. In fact, it may make the idea of a bundle easier to understand.However, we are primarily interested in vector bundles in this course, at least so much that we can understand the idea of the "derivative" in the category of smooth manifolds.

Note : It is a good idea to try solving exercises from [Dundas], they are generally simpler than those in [Barden&Thomas]. Both books have hints/solutions to the exercises, see for example chap.12 in [Dundas]|

 18 February On Monday 21.2 we shall discuss the topic in [Barden&Thomas] 1.4 and 1.5. The main goal is the embedding theorem for compact manifolds. In particular, see how bump functions are used to glue together locally defined smooth functions to obtain globally defined smooth functions. Dundas also gives a detailed discussion of these topics. We are still working with Chap.1 of [Barden&Thomas]. The material here is treated in more detail in Chap. 6 and 7 in [Dundas].Key concepts are : submanifold, immersion, embedding, submersion, inverse function theorem (and implicit function theorem, rank theorem, cf. [Dundas, Chap. 6], regular value, singular value, Whitney's imbedding theorem for compact manifolds, including bump functions needed to prove the theorem. After all this we will continue with Chap. 2 of [Barden&Thomas].Key concepts here are : the derivative of a smooth function, tangent vector, tangent space, tangent bundle (and vector bundle more generally). In [Dundas] this material is treated in Chap. 4 and 5, but the reader will see that two books have rather different approaches to the topic. [Dundas] is much more detailed, but we shall only explain the additional material from [Dundas] whenever needed.
 4 February Some of you may have the textbook [Barden&Thomas] now.The material of chap. 1 will be covered by the lectures. So Monday we continue with §1.3 about submanifolds,and §1.4, 1., leading to the Embedding Theorem. After that we continue with chap.2 which deals with the notion of derivative of smooth maps.