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Terilien
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What orginally motivated the field of differential geometry?
Beautiful shape of a potato !:rofl:Terilien said:What orginally motivated the field of differential geometry?
asub said: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.
asub said: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
pmb_phy said:asub said: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.
asub said:pmb_phy said: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.
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.asub said: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).
Hurkyl said: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.
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 said:In the last 10 years I've never seen Rn used for anything other than the collection of all n-tuples.
Are you pulling my leg?? let me tell you the first line in Schutz's math text that I mentioned above. From page 1Hurkyl said:The collection of all n-tuples is (usually) very uninteresting, and is almost never what's meant by R^n ...
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, ... , xn), 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.
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.Definition. If n is a positive integer, then the ordered-n-tuple is a sequence of real numbers (a1. a2, ... , an). The set of all ordered n-tuples is called n-space and is denoted by Rn
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 said: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 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.Hurkyl said: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.
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 said:The topology is established when you establish a metric.
That was the entire point I was trying to make. I guess it got lost in the noise.Hurkyl said: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.
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.Hurkyl said: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 ...
That wasn't an argument; it was a statement of fact.pmb_phy said:An argument that consists totally of "is almost never what's meant by ..." is hardly an argument at all.
The origin of differential geometry can be traced back to the work of ancient Greek mathematicians, such as Euclid and Pythagoras, who studied the properties of shapes and curves in two and three-dimensional space. However, the modern development of differential geometry is credited to the 18th and 19th century mathematicians, such as Leonhard Euler, Carl Friedrich Gauss, and Bernhard Riemann, who developed the concept of curved space and its applications to physics and mathematics.
One of the main motivations for the development of differential geometry was the need to understand and describe curved surfaces and spaces. This was important in the fields of astronomy, physics, and engineering, where curved surfaces and spaces were encountered. Another motivation was to generalize the concepts of geometry and calculus to curved spaces, which led to the development of new mathematical tools and techniques.
Differential geometry has numerous applications in various fields, including physics, engineering, computer graphics, and robotics. It is used to model and analyze physical systems with curved surfaces, such as the motion of planets and satellites in space. It is also used in the design of curved structures, such as bridges and buildings, and in the development of computer algorithms for creating and manipulating 3D objects.
Differential geometry is closely related to other branches of mathematics, such as calculus, algebra, and topology. It uses concepts and techniques from these fields to study the properties of curved spaces and surfaces. In particular, it is closely connected to differential calculus, which is used to study the behavior of functions on curved spaces.
There are several open problems in differential geometry, including the classification of all possible curved spaces and surfaces, the study of their intrinsic properties, and the development of new techniques for solving differential equations on curved spaces. Another open problem is the reconciliation of differential geometry with quantum mechanics, which is an ongoing topic of research in theoretical physics.