## 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