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Lecture 1: Basic definition of Manifold

(For simplicity, we will only consider the case)

 

Def 1: By , we mean the k-tuples () where .

 

Def 2: By open set, we mean the usual Euclidean open set.

 

Basic Ideas: A “manifold” is an object, which is “similar” to a normal object. Usually, A is “similar” to B if there exists a “specific” map f from A to B.

 

Def 3.1: A map f: U, where U are open, is smooth iff: all its partial derivative exists and continuous.

 

Ex 1:  f(x,y)=   where x,y are real numbers. So, f map from R.

 

Def 3.2: A map f:Uis smooth, U iff:

              for each x, there exist an open X containing x and F: such that .

 

Ex 2:  f(x)= where x lies in [0,1].

 

Homework 1.1: if f, g is smooth, gof is also smooth, provided that gf is well-defined. (Answer)

 

Def 4: A  map f  is a diffeomorphism iff f is a homeomorphism and f,are smooth.

 

Homework 1.2: Give an example of differomorphism and a proof. (Answer)

 

Def 5: A subset M in  is called a smooth manifold of dim m iff:

          For each x in M, there exist a nbhd and a diffeomorphism f mapping from to an open set V in

 

Graphic example of manifold:

 

                       

                  A sphere                                              A Torus