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## Main Question or Discussion Point

What orginally motivated the field of differential geometry?

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What orginally motivated the field of differential geometry?

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Hurkyl

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Beautiful shape of a potato !:rofl:What orginally motivated the field of differential geometry?

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At least, that is my understanding.

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JasonRox

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I didn't really know that. Thanks!

At least, that is my understanding.

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

The difference between R^{n}and E^{n}is that E^{n}is what you get when you add a metric to the space.

Pete

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mathwonk

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

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quasar987

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Hey bizzaro wonk, long time no see.

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mathwonk

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haaahaaahaa

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

The difference between R^{n}and E^{n}is that E^{n}is what you get when you add a metric to the space.

Pete

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.

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JasonRox

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

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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.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).

Here are a few select sources -

Pete

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Hurkyl

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It's been well over 15 years since I took abstract and linear algebra. In the last 10 years I've only seen RRis a highly overloaded symbol. It is generally cumbersome to distinguish betweenRthe affine space,Rthe vector space,Rthe topological space,Rthe differentiable manifold,Rthe complete ordered field,Rthe set,Rthe Lie group, et cetera. So, we streamline our thinking by lettingRdenote any of those things that is appropriate at the time -- and only wait until such distinctions are necessary before fixing exactly what we mean byR.

Note to

Best wishes

Pete

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Hurkyl

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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 weIn the last 10 years I've never seen R^{n}used for anything other than the collection of all n-tuples.

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Are you pulling my leg?? let me tell you the first line in Schutz's math text that I mentioned above. From page 1The collection of all n-tuples is (usually) very uninteresting, and is almost never what's meant by R^n ...

As another example consider the textThe space R^{n}is the usualn-dimensional space of vector algebra: a point in R^{n}is a sequence ofnreal numbers (x_{1}, x_{2}, ... , x_{n}), also called ann-tupleof real numbers. Intuitively we have the idea that this is acontinuousspace, that there are points of R^{n}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 R^{n}.

I take these identical definitions of RDefinition. Ifnis a positive integer, then theis a sequence of real numbers (aordered-n-tuple_{1}. a_{2}, ... , a_{n}). The set of all orderedn-tuples is calledand is denoted by Rn-space^{n}

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.

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Hurkyl

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

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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.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 settogether withthis additional structure is what he denotes as R^{n}.

Pete

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Hurkyl

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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 pointsThe topology is established when you establish a metric.

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That was the entire point I was trying to make. I guess it got lost in the noise.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 pointstogether withadditional structure -- in this case, a metric.

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.

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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.''

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mathwonk

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

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You're arguement 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 RThe 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 wereallythink 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 ...

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

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mathwonk

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Hurkyl

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That wasn't an argument; it was a statement of fact.An arguement that consists totally of "is almost never what's meant by ..." is hardly an arguement at all.

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.

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