FIRST MATH POST!

This ones for Zach and Steven.

So I have been bothered that my blog is not legitimate. For where is the math?

There are a thousand ways to answer this question, or at least I feel as though there are as I don't know how to truly answer it. For truth I never envisioned this page to be at all mathematical, or to discuss mathematical topics. It was more to act as a release from such times when I felt overwhelmed by mathematics, a place for "new ideas" so to speak. It is a hobby, an outlet for appreciation of the bands that get me through the day.
I could go on and on about the inspiration I feel at times listening to the music I choose to post, but usually with too much to say and little confidence to say it (story of my life?) I write very little, and again that will be the road I choose to take.

Now, that was all neither here-nor-there so back to the topic at hand:
Mathematics.
For the satisfaction mainly of the majority of the people I have coaxed into viewing my blog, those with an esteem for the music I post lying within any given epsilon of none (ie 90 percent of my hits thus far) and for the curiosity of those outside this circle from which I rarely venture. I will make a short post about a certain mathematical fascination of mine; Differential Geometry. (sorry but no PDE's this time)

As my lecture notes from today's class speak loosely:
"Differential Geometry is the study of Smooth Manifolds and special "geometric structures" on them.
Examples of such include: Riemannian Metrics and Simplectic Forms:

Now, if your first question is timidly; "What is a smooth manifold?" then I must say welcome to the world of terse, complicated mathematical definitions which are composed of in their brevity only an onslaught of more terse complicated definitions seemingly ad infinitum. (note: this is nearly always the case or in certain special circumstances -usually in diff geo- where the definitions in their excruciating detail take around a page to write out.)

If on the other hand your thoughts are more aligned with the phrase "Yes, I know. What then?"
Then the following links may be of some interest.

Futher Reading
Wikipedia!

But whatever the opinion of the esoteric definition above, if it struck fascination at all, I hope (at least from a mathematical perspective) that there is at least some vestige of the questions "Why is this important (what does this tell us)?" "What does this lead to (what does this infer)?" and "What does this mean?"
For if only platonic, there still must be a significance to a definition or result mathematically. "Ideas are only as good as the actions that follow and definitions are only as good as the results they lead to," to paraphrase my friend Zach.

I understand mathematics is not everyone's passion, at times I am even unsure if it is mine -it is certainly true that specific realms are not- so I have not intended upon being overbearing, and I apologize if I have.

"We can forgive a man for making a useful thing as long as he does not admire it. The only excuse for making a useless thing is that one adores it intensely" - Oscar Wilde

I will end this now as not burden you with any more of my ramblings. Feel free to comment and tear me up for offending you as well.