Riemann Geometry: Where is the Flaw in My Thinking?

[Gear300]

One of the axioms of Riemann's geometry holds that there are no parallel lines and that any two lines meet. Since Riemann's geometry fits for that of a sphere, any two great circles of the sphere should intersect. However, if we were to take 2 longitudinal lines, then it is possible that these lines never meet. Where is the flaw in my thinking?

 

[Gear300]

Never mind...I overlooked how the longitudinal lines are not by definition "straight" lines.

 

[HallsofIvy]

Specifically, they aren't great circles.

 

[Office_Shredder]

You mean latitude?

 

[HallsofIvy]

You mean latitude?

Good point. "Longitudinal lines" isn't clear but since he referred to them never meeting, I assumed that was what he meant. "Lines of constant longitude", of course, meet at the north and south poles.

 

[HallsofIvy]

By the way, back many many years ago, when I was in high school, I asked my geometry teacher, "why not just define 'parallel lines'" as being lines a constant distance apart? Then it would be obvious that they don't meet and there is only one 'parallel' to a given line through a given point". Being a high school teacher he pretty much just brushed off the question. Now I know that you need the parallel postulate to prove that the set of points at constant distance equidistant from a given line is a "line". The set of lines of lattitude are examples of "equidistant curves" on the spere.