(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