Original motivation of differential geometry

[Terilien]

What orginally motivated the field of differential geometry?

 

[Hurkyl]

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.

 

[tehno]

What orginally motivated the field of differential geometry?

Beautiful shape of a potato !:rofl:

 

[asub]

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 high school 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.

 

[JasonRox]

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 high school 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!

 

[pmb_phy]

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).
[/quite]It appears that you may be off as to the definition of affine space? Please see - http://mathworld.wolfram.com/AffineSpace.html

The difference between Rⁿ and Eⁿ is that Eⁿ is what you get when you add a metric to the space.


Pete

 

[mathwonk]

money, fame, sex. these are the prime motivators.

 

[quasar987]

Hey bizzaro wonk, long time no see.

 

[mathwonk]

haaahaaahaa

 

[asub]

It appears that you may be off as to the definition of affine space? Please see - http://mathworld.wolfram.com/AffineSpace.html

The difference between Rⁿ and Eⁿ is that Eⁿ is what you get when you add a metric to the space.


Pete

E^n is not what you add a metric to R^n. Please see Boothby (2ed, p. 4) for a discussion of this (he says that most texts give the same incorrect definition that you are using).

An affine space is a space without the origin, but with most of the nice properties of vector spaces. In particular, you cannot add points in an affine space. I don't see how Mathworld and I disagree.

 

[JasonRox]

E^n is not what you add a metric to R^n. Please see Boothby (2ed, p. 4) for a discussion of this (he says that most texts give the same incorrect definition that you are using).

An affine space is a space without the origin, but with most of the nice properties of vector spaces. In particular, you cannot add points in an affine space. I don't see how Mathworld and I disagree.

It's what my prof. described to me. Exactly what you're saying.

 

[pmb_phy]

E^n is not what you add a metric to R^n. Please see Boothby (2ed, p. 4) for a discussion of this (he says that most texts give the same incorrect definition that you are using).

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

Topology, Dugundji

Pete

 

[Hurkyl]

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.

 

[pmb_phy]

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.

Best wishes
Pete

 

[Hurkyl]

In the last 10 years I've never seen Rⁿ used for anything other than the collection of all n-tuples.

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 ...

 

[pmb_phy]

The collection of all n-tuples is (usually) very uninteresting, and is almost never what's meant by R^n ...

Are you pulling my leg?? let me tell you the first line in Schutz's math text that I mentioned above. From page 1

The space Rⁿ is the usual n-dimensional space of vector algebra: a point in Rⁿ is a sequence of n real numbers (x₁, x₂, ... , x_n), also called an n-tuple of real numbers. Intuitively we have the idea that this is a continuous space, that there are points of Rⁿ arbitrarily close to any given point, that a line joining any two points can be subdivided into arbitrarily many pieces that also join points in Rⁿ.

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

Definition. If n is a positive integer, then the ordered-n-tuple is a sequence of real numbers (a₁. a₂, ... , a_n). The set of all ordered n-tuples is called n-space and is denoted by Rⁿ

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 textbook The Meaning of Relativity.

Pete

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.

 

[Hurkyl]

Are you pulling my leg?? let me tell you the first line in Schutz's math text that I mentioned above. From page 1

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.

 

[pmb_phy]

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.

Pete

 

[Hurkyl]

The topology is established when you establish a metric.

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.

 

[pmb_phy]

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.

Best regards

Pete

ps - Since I really hate debating definitions I won't be posting any more in this thread. Thanks everyone for your thoughts.

 

[asub]

Hi pmb_phy, sorry for being pedantic, but I guess this thread is about being pedantic :)

When I was pointing out Boothby's book, I was not pointing out his definition of Euclidean space--after all it was defined by Euclid. And Euclid's definition of E^2 has nothing to do with numbers or ordered pairs. It is only concerned about points, lines, triangles, parallelograms, etc. and proving their properties without the use of coordinates. Boothby gives a discussion and some historical perspective about this issue in the chapter.

My main gripe is not that R^n is sometimes used as E^n--they are just symbols. My problem is with the idea that a vector space with metric defined by n-tuple of numbers is equivalent to a Euclidean space. First of all, Euclidean space has no axes. And the vector space has no objects such as lines, planes, spheres, etc.

I will quote few sentences from Boothby:

``The identification of R^n and E^n came about after the invention of analytic geometry by Fermat and Decartes and was eagerly seized upon since it is very tricky and difficult to give a suitable definition of Euclidean space, of any dimension, in the spirit of Eulid, that is, by giving axioms for (abstract) Euclidean space as one does for abstract vector spaces. This difficulty was certainly recognized for a very long time, and has interested many great mathematicians.[...]A careful axiomatic definition of Euclidean space is given by Hilbert.[...] It is the existence of such coordinate mappings which make the identification of E^2 and R^2 possible. But caution! An arbitrary choice of coordinates is involved, there is no natural, geometrically determined way to identify the two spaces. Thus at best, we can say that R^2 may be identified with E^2 plus a coordinate system.''

 

[mathwonk]

hmmmmmmm...

"inky dinky doo, was the highly interesting song that he sang.." humphrey bogart.

 

[pmb_phy]

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 ...

You're argument consists entirely of definitions that I have no access to. My references agree with what I've posted as far as definitions go and that includes what Rⁿ. An argument that consists totally of "is almost never what's meant by ..." is hardly an argument at all. The rest is beyond my recollection since its been close to two decades since I've studied this material other than that found in tensor analysis books like Schutz etc.

Since I've reached a point of saturation where I can't see of me posting of anything more than I already have I will not post again in this thread and will respond only in PM.

Thank you.

Pete

 

[mathwonk]

pete, i think you and hurkyl are just speaking a different vocabulary here, you are both right, but not meshing.

 

[Hurkyl]

An argument that consists totally of "is almost never what's meant by ..." is hardly an argument at all.

That wasn't an argument; it was a statement of fact.

Frankly, I'm confused as to what point you're trying to make. In #20 you seem to agree that R^n (generally) isn't used to refer to a mere set, but instead to refer to a set together with additional structure, so it's unclear why you appear to be opposing my point.My best guess (which is based on trying to figure out why you're arguing, rather than the content of your posts) is that you are trying to insist that people only ever use R^n to refer to the thing Schutz defined, and they never use that symbol to refer to anything else. (Note that that quote doesn't even say if he's defining a vector space, a topological space, a differentiable manifold, or even a mere set of points. Not having the text, I can't look it up) I don't know what to say except that in my experience studying and practicing mathematics, I've seen the symbol used for whichever of those (closely related) structures is the kind of object under study. E.g. when doing set theory, one would use R to denote a certain set, and R^n to denote the set of n-tuples with components in R. But when doing linear algebra, one would use R to denote a certainl field, and R^n to denote the (usual) (real-)vector space structure on the set of n-tuples.

 

[Hurkyl]

Whoops, didn't mean to hit delete! One shouldn't moderate late at night. Thread restored.