Riemann Geometry: Where is the Flaw in My Thinking?

In summary, the conversation discusses the axiom of Riemann's geometry that states there are no parallel lines and any two lines meet. This is applicable to a sphere, where two great circles will intersect. However, there is a flaw in the thinking when considering 2 longitudinal lines, as they are not considered "straight" lines and thus will not intersect. The flaw is corrected by clarifying that "longitudinal lines" actually refers to "lines of constant longitude", which do intersect at the north and south poles. The conversation also mentions a question about defining "parallel lines" as lines that are a constant distance apart, but it is explained that the parallel postulate is needed to prove this concept. The set of lines of latitude on
  • #1
Gear300
1,213
9
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?
 
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  • #2
Never mind...I overlooked how the longitudinal lines are not by definition "straight" lines.
 
  • #3
Specifically, they aren't great circles.
 
  • #4
You mean latitude?
 
  • #5
Office_Shredder said:
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.
 
  • #6
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.
 

FAQ: Riemann Geometry: Where is the Flaw in My Thinking?

What is Riemann Geometry?

Riemann Geometry is a branch of mathematics that studies curved surfaces and spaces. It is named after mathematician Bernhard Riemann and is a fundamental concept in the field of differential geometry.

What is the significance of Riemann Geometry?

Riemann Geometry is significant because it provides a mathematical framework for understanding and describing the physical world, particularly in the field of General Relativity. It is also used in various other fields such as physics, engineering, and computer science.

What is the flaw in my thinking regarding Riemann Geometry?

Without more specific information about your thinking, it is difficult to identify a specific flaw. However, Riemann Geometry is a complex subject and it is common for individuals to have misconceptions or misunderstandings. It is important to carefully study and understand the principles of Riemann Geometry to avoid potential flaws in thinking.

How is Riemann Geometry applied in real-world scenarios?

Riemann Geometry has numerous applications in the real world, such as in the study of curved surfaces in physics and engineering, in the development of computer graphics and animation, and in understanding the shape and dynamics of the universe. It is also used in various fields of research, including geology, biology, and economics.

What are some useful resources for learning more about Riemann Geometry?

There are many resources available for learning about Riemann Geometry, including textbooks, online lectures, and tutorials. Some popular textbooks include "Riemannian Geometry" by Manfredo do Carmo and "Differential Geometry: Curves, Surfaces, Manifolds" by Wolfgang Kühnel. Online resources such as Khan Academy and MIT OpenCourseWare also offer free courses and lectures on Riemann Geometry.

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