What orginally motivated the field of differential geometry?
I'm not sure, but I think it was the desire to study the "intrinsic" geometry of surfaces -- e.g. to study a sphere as an object in its own right, rather than as a subset of Euclidean 3-space.
Beautiful shape of a potato !:rofl:
I think the motivation for differential geometry becomes clear when one stops indentifying R^n with E^n (incorrectly). Euclidean space is defined by a set of axioms and is actually not even a vector space (it's affine space). Some time between highschool and college most people identify R^n with E^n and take differentiation and integration in E^n for granted. What we are actually doing is identifying inner product space R^n with E^n and a coordinate system. So, if we want to do differentiation and integration in a curved space we identify a neighbourhood of the curved space and a map with with the inner product space R^n. The curved space with all such maps is the manifold. Differential geometry just involves studying how to do the familiar differentiation and integration in this new creature.
At least, that is my understanding.
I didn't really know that. Thanks!
money, fame, sex. these are the prime motivators.
Hey bizzaro wonk, long time no see.
Sorry but I don't have that text. The texts I do have contradict what you and your source(s) are trying to say. My sources (i.e. texts/other physicists) are differential geometry texts, GR texts and I just got a new text on topology which aslo agrees with my other sources.
Here are a few select sources -
Gravitation, Misner, Thorne and Wheeler
Geometrical Methods of Mathematical Physics, Bernard F. Schutz
R is a highly overloaded symbol. It is generally cumbersome to distinguish between R the affine space, R the vector space, R the topological space, R the differentiable manifold, R the complete ordered field, R the set, R the Lie group, et cetera. So, we streamline our thinking by letting R denote any of those things that is appropriate at the time -- and only wait until such distinctions are necessary before fixing exactly what we mean by R.
It's been well over 15 years since I took abstract and linear algebra. In the last 10 years I've only seen Rn used for to mean the collection of all n-tuples. If the symbol is used in other places besides differential geometry and tensor analysis then I have no recollection of it. I have a few abstract algebra texts so I'll take a gander. But I doubt that all possible groups have a metric and thus doesn't apply in the sense we've been talking about. Perhaps I'll learn something new today by doing this. :)
Note to asub: I wanted to make a point, the main purpose being that I don't want to come across as being "stuffy" or whatever. I'm not sure how I came across in my post to you. What I had neglected to mention was a very important fact: Because one author says that other authors define something incorrectly can only be taken to mean that the authors disagree on the definition. It cannot be taken to mean that one is right and the other wrong. One may be more popular than another. One may be so unpopular that it doesn't get past the editor. But the way I've always interpreted this has come from my GR professor, GR experts that I know and, as I mentiond before, the texts that I read. I'm also willinbg to scan in the page of any text I referenced so that you can read exactly what I read and in the context I read it. Upon yours or someone elses request that is.
The collection of all n-tuples is (usually) very uninteresting, and is almost never what's meant by R^n -- that is merely the underlying set for the things we really think are interesting, like R^n the differentiable manifold, or R^n the topological space, or R^n the vector space, or R^n the commutative ring, or ...
Are you pulling my leg?? let me tell you the first line in Schutz's math text that I mentioned above. From page 1
As another example consider the text Elementary Linear Algebra, by Howard Anton. In sectoipon 4.1 "Euclidean n-space[/i]" which reads on page 133
I take these identical definitions of Rn as the definition of Rn, especially since it agrees with every other text I have (except for one). Even Einstein defined Euclidean space in this manner in his renowned text book The Meaning of Relativity.
ps - To all posters - If you have a solid reference to a text that I have then please let me know so that you can send or post exactly what it states on this. You've got me curious now.
And right there in the first page, we see Schutz not talking merely about a set of n-tuples, but also of a topology (close to) on that set of n-tuples, and an affine structure (line) on that set of n-tuples. This set together with this additional structure is what he denotes as Rn.
The topology is established when you establish a metric. Why you mention a line is beyond me. Schutz was merely using it to make a point about closeness of points.
Sure, but the collection of n-tuples doesn't have a metric. It's just a collection of n-tuples. A metric space is a set of points together with additional structure -- in this case, a metric.
That was the entire point I was trying to make. I guess it got lost in the noise.
ps - Since I really hate debating definitions I won't be posting any more in this thread. Thanks everyone for your thoughts.
Separate names with a comma.