- #1

- 56

- 0

## Homework Statement

Given S

^{1}={(x,y) in R

^{2}: x

^{2}+y

^{2}=1}. Show that S

^{1}is a 1-dimensional manifold.

## Homework Equations

## The Attempt at a Solution

Let f

_{1}:(-1,1)->S

^{1}s.t. f

_{1}(x)=(x,(1-x

^{2})

^{1/2}).

This mapping is a diffeomorphism from (-1,1) onto the top half of the circle S

^{1}.

I was trying to write a prove that f

_{1}is indeed a diffeomorphism, but I am having trouble showing the onto part.

I argued that f

_{1}is onto by defining the inverse map and showing that the inverse map is 1 to 1 and hence f

_{1}is onto.