Riemann Geometry

  • Thread starter Gear300
  • Start date
1,166
5
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?
 
1,166
5
Never mind...I overlooked how the longitudinal lines are not by definition "straight" lines.
 

HallsofIvy

Science Advisor
Homework Helper
41,682
864
Specifically, they aren't great circles.
 

Office_Shredder

Staff Emeritus
Science Advisor
Gold Member
3,734
98
You mean latitude?
 

HallsofIvy

Science Advisor
Homework Helper
41,682
864
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

Science Advisor
Homework Helper
41,682
864
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.
 

Related Threads for: Riemann Geometry

  • Posted
Replies
6
Views
814
  • Posted
Replies
5
Views
3K
Replies
1
Views
546
D
  • Posted
Replies
1
Views
2K
  • Posted
Replies
5
Views
3K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Hot Threads

Top