# Enveloping spaces

1. Nov 3, 2003

How do you draw a curve with line element ds2=y2dx2+dy2 in a R2 space? Is it just lots of x=a lines, with a-any real number?

I don't understand why a mapping of phi(x,y)=(x/y,sqrt(x2+y2)) can be an enveloping space for the above curve?

Any ideas, anyone?

2. Nov 17, 2003

### lethe

this doesn t quite make sense. if you draw a curve in R2, it inherits an induced metric from the normal euclidean metric.

this metric is
$$ds = \sqrt{1+\left(\frac{df}{dx}\right)^2}dx$$
from the euclidean metric
$$ds^2 = dx^2 + dy^2$$

i cannot get the line element you wrote from a curve in euclidean R2
what is an enveloping space? i have never heard this term before. can you define it please?

3. Nov 21, 2003

It's from this paper gr-qc/9405063.

Can I substitute make x'=arctan(y/x) and y'=sqrt(x2+y2) into ds2=dx2+dy2 so that phi'(x,y)=(x',y') be the covering space for the original Euclidean space with (0,0) removed?