Enveloping spaces

[maddy]

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?

[lethe]

Originally posted by maddy
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?
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
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?

what is an enveloping space? i have never heard this term before. can you define it please?

[maddy]

what is an enveloping space? i have never heard this term before. can you define it please?

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

this doesn't quite make sense. if you draw a curve in R2, it inherits an induced metric from the normal euclidean metric.

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?