The discussion centers around the study of Riemannian geometry, particularly whether someone who has completed high school can begin to learn about it, including its relevance to astronomy. Participants explore the prerequisites and resources needed for understanding this branch of mathematics.
Participants generally agree on the need for a strong mathematical foundation to study Riemannian geometry, but there is no consensus on the best approach or resources, as multiple suggestions are offered without resolution.
Some participants mention specific prerequisites and resources, but there is no agreement on a single path to learning Riemannian geometry, indicating a variety of approaches may be necessary depending on individual backgrounds.
jedishrfu said:You can read about it at Wikipedia:
https://en.wikipedia.org/wiki/Riemannian_geometry
Its a branch of Differential Geometry. To fully understand it you'll need some background in Calculus, Linear Algebra, Differential Equations and Vector/Tensor Analysis.
jedishrfu said:To get an understanding of some of the topics study the videos at:
www.MathIsPower4U.com
They cover topics from HS to Calculus I II and III,Linear Algebra and Differential Equations and some Vector Analysis in Calculus III.
For the other stuff you'll need textbooks. However the MIT Online Courses may cover Tensor Analysis and Differential Forms and even Differential Geometry itself.
Here's some of the Differential Geometry courses they have:
http://search.mit.edu/search?site=ocw&client=mit&getfields=*&output=xml_no_dtd&proxystylesheet=http://ocw.mit.edu/search/google-ocw.xsl&requiredfields=WT%2Ecg_s:Course+Home|WT%2Ecg_s:Resource+Home§ionlimit=WT%2Ecg_s:Course+Home|WT%2Ecg_s:Resource+Home&as_dt=i&oe=utf-8&departmentName=web&filter=0&courseName=&q=differential+geometry&btnG.x=0&btnG.y=0
I'll leave it to you to search the site for the others I mentioned earlier.
chiro said:Hey jamalkoiyess.
Riemannian geometry is basically looking at geometry where the different components (like axis, basis vectors, etc) are dependent.
This means that instead of different pieces of information being independent (where you change one piece of information and the others stay the same), they are dependent (meaning you change one piece of information and it will - in some case change something else).
That is the intuition behind differential and Riemannian geometry. It just means that instead of the information in each co-ordinate (or basis vector) being independent, they relate to each other in some way.
Things like space-time have this property where if you change one thing then it changes the kinds of values that the other should have.
It used to be that people thought that the best way to understand things was through independence - but eventually a few mathematicians realized that it might be a good idea to look past that and it's applications have been used extensively (including in the well known use of General Relativity).