Site hosted by Build your free website today!

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