Curvature of Non-Euclidean Space

Date: 05/22/2000 at 11:44:09
From: Michael Rosenof
Subject: Positive/Negative Curvature

What is the difference between positive and negative curvature in 
non-Euclidean geometry? I looked through your archives and I couldn't 
find the answer.

Thank you.

Date: 05/22/2000 at 15:58:27
From: Doctor Rob
Subject: Re: Positive/Negative Curvature

Thanks for writing to Ask Dr. Math, Michael.

In a space of positive curvature, like the surface of a sphere, there 
are no lines parallel to a given line through an outside point.

In a space of negative curvature, like the surface of an hyperbolic 
paraboloid, there are multiple lines parallel to a given line through 
an outside point.

In a space of zero curvature, like a plane, there is a single line 
parallel to a given line through an outside point.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/