# Riemann Geometry

#### 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

Homework Helper
Specifically, they aren't great circles.

#### Office_Shredder

Staff Emeritus
Gold Member
You mean latitude?

#### HallsofIvy

Homework Helper
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

Homework Helper
By the way, back many many years ago, when I was in highschool, 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.

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