Ways to abstract curves or surfaces

  • #1
Hello. So, we have curves and surfaces. We already know about generally manifolds and Riemannian manifolds but what i want to produce are ways to abstract curves or surfaces but i am not talking about manifolds. Do you have any ideas? Perhaps the feature of curvature would help? To make an abstraction of curvature and define the corresponding objects? We know about the Gaussian curvature, so how would someone abstract it? Thank you.
 

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  • #2
martinbn
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Hello. So, we have curves and surfaces. We already know about generally manifolds and Riemannian manifolds but what i want to produce are ways to abstract curves or surfaces but i am not talking about manifolds. Do you have any ideas? Perhaps the feature of curvature would help? To make an abstraction of curvature and define the corresponding objects? We know about the Gaussian curvature, so how would someone abstract it? Thank you.
So, you want to do something, but you don't know what, and you are asking for ideas!!!
 
  • #3
So, you want to do something, but you don't know what, and you are asking for ideas!!!
Ok, i am sorry if what i read is offensive, i wanted a discussion to understand and if something like that already exists. Perhaps i should not write it, because it might sound like i want to steal ideas.
 
  • #4
Perhaps the thread should be deleted?
 
  • #5
martinbn
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Ok, i am sorry if what i read is offensive, i wanted a discussion to understand and if something like that already exists. Perhaps i should not write it, because it might sound like i want to steal ideas.
No, I don't think you are trying to steal ideas. But it is not clear what you are asking for, so how can anyone help you?
 
  • #6
Are forums the right place or not for collaboration in producing original works in sciences? I should be more careful from now on when writing in forums i think.

What i am trying to do is generalise the Gaussian curvature for the right kind of geometric sets.
 
  • #7
If someone generalises the Gaussian curvature and places it on the right abstract geometric sets he gets different generalised geometric sets.

I think someone needs a second or perhaps a third also example to make the generalisation.
 
  • #8
martinbn
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Are forums the right place or not for collaboration in producing original works in sciences? I should be more careful from now on when writing in forums i think.
I suppose forums could in principle be used for that.
What i am trying to do is generalise the Gaussian curvature for the right kind of geometric sets.
What are the right kind of geometrical sets? This is not how original work is done by the way. You don't just try to randomly generalize something. You probably don't have experience in doing research. It would be best to go through the usual process. Get the education, work with an advisor, and then you can do research.
 
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  • #9
I suppose forums could in principle be used for that.

What are the right kind of geometrical sets? This is not how original work is done by the way. You don't just try to randomly generalize something. You probably don't have experience in doing research. It would be best to go through the usual process. Get the education, work with an advisor, and then you can do research.
Yes, indeed i do not have experience in doing research. I study at a university at a maths department.I want to get the degree but also i am interested in doing scientific research although i do not have a doctorate degree.
 
  • #10
I got stuck at my courses at university. I am too afraid to give exams and pass the tests. What should i do? Also, at the problem solving processes i do not know what to do many times.
 
  • #11
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Manifolds are already a generalization. They are as abstract as it can get. Another approach is via algebraic varieties. The main problem with your question is: What is a surface or curve? Try to define it and you will end up with a manifold, if you want to do analysis, and with an algebraic variety if you want to inspect components, intersections and zeros.
 
  • #12
Perhaps trying to pass courses at university without reading from the notes of the teacher is very difficult? Or without trying to solve exercises from those notes?
 
  • #13
Manifolds are already a generalization. They are as abstract as it can get. Another approach is via algebraic varieties. The main problem with your question is: What is a surface or curve? Try to define it and you will end up with a manifold, if you want to do analysis, and with an algebraic variety if you want to inspect components, intersections and zeros.
I think they have different definitions in the case of differential geometry and in that of algebraic geometry and yes in those cases so far the generalisation was that of manifolds and algebraic varieties.
 
  • #14
Different definitions of curves or surfaces could produce different generalisations?
 
  • #16
Please define your surface.
In differential geometry a surface is a two dimensional differentiable manifold with a metric tensor i think.In algebraic geometry it is a polynomial equation of three variables i think. I do not yet know how to define the surface in different ways.
 
  • #17
Perhaps a different definition of an n dmensional differentiable manifold with a metric tensor is needed not that of a surface to make a generalisation of a Riemannian manifold.
 
  • #18
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If you define a manifold with a manifold, then we still do not know what you mean. What is a curve? Define it without using other geometric terms, for otherwise you will only circle around it.
 
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  • #19
Perhaps the thread should be closed, i do not know the answer yet.
 
  • #20
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Perhaps the thread should be closed, i do not know the answer yet.
I think that you try to reinvent the wheel. When I was a kid I tried to figure out whether there could be a binary operation ##\circ## such that ##\log(a+b)=\log(a)\circ \log (b)##. I was fascinated by the fact that we can turn powers into multiplications ##\log(a^b)=b\log(a)## and multiplications into additions ##\log(a\cdot b)=\log(a)+\log(b)##. So why not turn additions into something even smaller? I started to think about the fact that we have a fixed point: ##2^2=2\cdot 2 = 2+2=4,## so my only requirement was ##2\circ 2 = 4.##

Now, many years later, I know that there cannot be such an operation. Why? Because of the Leibniz rule of differentiation, or even more general: because of the definition of derivations. Differentiation of products yield sums, but differentiation of linear functions are themselves linear functions. There is no smaller binary operation than addition. However, this insight took me a complete study of mathematics and time to gain an overview of the underlying principles. As tempting as it might be to revolutionize a method, as it is in vain. To see this may take a while, and I have found other interesting unknown concepts. There are no shortcuts in science. You will have to do it like everybody else: study, learn, talk a lot about it, read even more, and maybe then, but surely not before then, you may find new concepts and views.
 
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