February 12 Discussion

For those who are not sure what Riemannian geometry is (including myself), there is a link that I think really helps, it designed for nonmathematician, so I am pretty sure everyone is going to get it, so read it before today's class if you want to.

Lin


This is a link that may be helpful/interesting, it is an english translation of one of Poincare's articles about Non-Euclidean geometry. It contains a commentary by Poincare about the work of Euclid, Lobachevsky and Riemann.

http://www-history.mcs.st-and.ac.uk/Extras/Poincare_non-Euclidean.html
-Angela Rogers


I wanted to investigate Poincare and Riemannian Geometry further, as these were concepts that were either confusing or had not been introduced but seemed interesting.

I found a really nice summary of some of Saccheri's work as well as some of Poincare's. It appears to be done by someone at the University of Kentucky. Click here if you wish to take a look. It helped me by giving me another summary of their work, along with some visual representations, which have been helpful in trying to understand some of these concepts.

In browsing, I also found an interesting PDF of what appeared to be lecture notes detailing a more calculus-based and computationally rigorous approach than we have been taking. It has an interesting visual of the unit disk on page four and can be accessed by clicking here.

Finally, I found a really interesting website in which the designer has implemented a hands-on poincare disk using what looks like Adobe Flash. You can take a look here.

I then turned to Riemmannian Geometry, seeking a better explanation of what it was an its implications on the history of mathematics. As I typically do, I took a look at what Wikipedia had to say, and it was somewhat intimidating. You all might find it as a good place to start, though some may feel, as do I, that Wikipedia may get fairly technical with complex terms and concepts, which may make understanding such terms and concepts more difficult.

I then looked at a scholarly paper, thinking it would give me more background. This paper was also very intensive, and covered a lot of concepts in topology. While I understood some of these concepts in the paper, such as tangent vectors, they were presented in a different way here that made it more confusing. It does not appear that Riemann appears in the chapter we are on in the textbook, but I did look him up in the index and find other interesting sections of the textbook in which Riemann is discussed. I would be interested to learn more about Riemannian Geometry from a more digestible approach, as I feel it would give me a good perspective.

—Rob Moray

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