Topology vs Differential Geometry

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Hello. I am studying Analysis on Manifolds by Munkres. My aim is to be able to study by myself Spivak's Differential Geometry books. The problems is that the proof in Analysis on Manifolds seem many times difficult to understand and I am having SERIOUS trouble picturing myself coming up with some of them (most of them). Especially those that use so many topology stuff (like cubes that cover othter rectanlge which are of measure 0 and they are included in a sequence of compact sets such that Cn is in Cn+1 for each n ....... CHAOS). Do you think that studying Munkres' Topology book instead would be more benefiacial? Which would help more in physics or in being more proficient in mathematics (for the puproses of physics)? Is topology in general helpful in physics and where? Thank you in advance for your answers!
 

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  • #2
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Hello. I am studying Analysis on Manifolds by Munkres. My aim is to be able to study by myself Spivak's Differential Geometry books. The problems is that the proof in Analysis on Manifolds seem many times difficult to understand and I am having SERIOUS trouble picturing myself coming up with some of them (most of them). Especially those that use so many topology stuff (like cubes that cover othter rectanlge which are of measure 0 and they are included in a sequence of compact sets such that Cn is in Cn+1 for each n ....... CHAOS). Do you think that studying Munkres' Topology book instead would be more benefiacial? Which would help more in physics or in being more proficient in mathematics (for the puproses of physics)? Is topology in general helpful in physics and where? Thank you in advance for your answers!
In my opinion you won't need to concentrate on topology. I think the fundamental terms and relations are sufficient, as these things you mentioned probably won't be part of a textbook about topology. Analysis on manifolds and differential geometry are closely related and both far more important to physics, as topology is - again my opinion. A topologist might have a different point of view. I would rather try and draw much more of what you think you have difficulties with. It's often a basic principle which has a correspondence in low dimensional spaces. You shouldn't draw an accurate image since you aren't planning to construct a building, but a few sketches can be very helpful. So my answer to your questions is:
  • No.
  • Stay with the two books you mentioned (Analysis on Manifolds, Differential Geometry)
  • Try to sketch situations and "see" what's going on in the proofs.
  • Come on over and ask specific questions here on PF. It will require some effort to show them to us, but that's still better than to get stuck or learn something wrong. It also will help you top memorize if you are forced to explain them to others.
  • Another topic - topology - is a good idea on its own, however, not necessarily helpful for the above, which touches only a small section of topology.
 
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  • #3
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In my opinion you won't need to concentrate on topology. I think the fundamental terms and relations are sufficient, as these things you mentioned probably won't be part of a textbook about topology. Analysis on manifolds and differential geometry are closely related and both far more important to physics, as topology is - again my opinion. A topologist might have a different point of view. I would rather try and draw much more of what you think you have difficulties with. It's often a basic principle which has a correspondence in low dimensional spaces. You shouldn't draw an accurate image since you aren't planning to construct a building, but a few sketches can be very helpful. So my answer to your questions is:
  • No.
  • Stay with the two books you mentioned (Analysis on Manifolds, Differential Geometry)
  • Try to sketch situations and "see" what's going on in the proofs.
  • Come on over and ask specific questions here on PF. It will require some effort to show them to us, but that's still better than to get stuck or learn something wrong. It also will help you top memorize if you are forced to explain them to others.
  • Another topic - topology - is a good idea on its own, however, not necessarily helpful for the above, which touches only a small section of topology.
Where is topology useful in physics? I show some sections of Spivak's Differential Geometry book and Munkres' complicated proofs and it seemed topology is a really useful mathematical TOOL for other things. My problem is that I am probably going to specialize in particle physics, quantum theory and perhaps even string theory (if I find these interesting). Do you think that still Differential Geometry is better for me?
 
  • #5
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Where is topology useful in physics? I show some sections of Spivak's Differential Geometry book and Munkres' complicated proofs and it seemed topology is a really useful mathematical TOOL for other things.
Sure it is. The question is: how much of it? You will probably know what an open set is, a closed set, a covering, a continuous map, limit points, closures, completions or metrics, and maybe some other fundamental concepts. They are indeed needed and basic concepts which should be known. The question is whether you need a full understanding of the field beforehand? It might be the case that you will be confronted with simplicial complexes and cohomologies, but will there be a need to study them beforehand? And here's where I said no, I don't think so. It could as well be done on demand. Of course if you will specialize in certain fields, there will be further mathematical concepts needed. E.g. in physics you will mainly have spaces which allow a kind of metric. But metric spaces are only a small part of topology, so why learn what a Sierpinski space is? If it will occur, look it up. If you want to learn everything in mathematics which might be useful in physics, well, then you probably have to study both and good physicists are often good mathematicians, too. It's primarily the range of knowledge that differs.
My problem is that I am probably going to specialize in particle physics, quantum theory
Functional analysis (Operators and Hilbert spaces), differential geometry (which basically includes analysis on manifolds) (coordinate systems), linear algebra (basics), measure theory, resp. stochastic (Lebesgue integration, real (hyph.) analysis, probability theory).
and perhaps even string theory (if I find these interesting).
Abstract algebra in a very wide sense.
Do you think that still Differential Geometry is better for me?
Yes. Have a look on the list above. I think it is long enough to merely learn those topological concepts which arise within them, resp. the fundamentals, resp. to learn it on demand. Differential geometry is basically the complete physics: spacetime isn't Euclidean, everything is written in Lagrangians and differential equations, resp. differential coordinates, even classical physics as fluid mechanics. You meet its language all of the time, so the better you understand it the easier will be physics.

You mentioned string theory. This is based on the standard model, which is about Lie groups and their representations, ergo analytical groups, i.e. smooth manifolds. That they are also topological groups is just a side note. I do not claim you won't need to know what a couple of topological definitions are, e.g. the different versions of connectivity, compactness etc. I simply think that this can be learnt along the way or if you're a commuter, on the train. Buy (or download) a cheap paperback about topology and that should do. If you're done with differential geometry, you will automatically have a good basis of topology - at least the part which is used in physics. So the question is: look it up or study it in advance. In the end this is a matter of taste.
 
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  • #6
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Sure it is. The question is: how much of it? You will probably know what an open set is, a closed set, a covering, a continuous map, limit points, closures, completions or metrics, and maybe some other fundamental concepts. They are indeed needed and basic concepts which should be known. The question is whether you need a full understanding of the field beforehand? It might be the case that you will be confronted with simplicial complexes and cohomologies, but will there be a need to study them beforehand? And here's where I said no, I don't think so. It could as well be done on demand. Of course if you will specialize in certain fields, there will be further mathematical concepts needed. E.g. in physics you will mainly have spaces which allow a kind of metric. But metric spaces are only a small part of topology, so why learn what a Sierpinski space is? If it will occur, look it up. If you want to learn everything in mathematics which might be useful in physics, well, then you probably have to study both and good physicists are often good mathematicians, too. It's primarily the range of knowledge that differs.

Functional analysis (Operators and Hilbert spaces), differential geometry (which basically includes analysis on manifolds) (coordinate systems), linear algebra (basics), measure theory, resp. stochastic (Lebesgue integration, real (hyph.) analysis, probability theory).

Abstract algebra in a very wide sense.

Yes. Have a look on the list above. I think it is long enough to merely learn those topological concepts which arise within them, resp. the fundamentals, resp. to learn it on demand. Differential geometry is basically the complete physics: spacetime isn't Euclidean, everything is written in Lagrangians and differential equations, resp. differential coordinates, even classical physics as fluid mechanics. You meet its language all of the time, so the better you understand it the easier will be physics.

You mentioned string theory. This is based on the standard model, which is about Lie groups and their representations, ergo analytical groups, i.e. smooth manifolds. That they are also topological groups is just a side note. I do not claim you won't need to know what a couple of topological definitions are, e.g. the different versions of connectivity, compactness etc. I simply think that this can be learnt along the way or if you're a commuter, on the train. Buy (or download) a cheap paperback about topology and that should do. If you're done with differential geometry, you will automatically have a good basis of topology - at least the part which is used in physics. So the question is: look it up or study it in advance. In the end this is a matter of taste.
Thank you very much for your extensive reply!
 
  • #7
FactChecker
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I think that topology as a central concern of physics is limited to a very few specialized areas. Certainly manifolds and differential geometry are much more applicable. IMHO, the relevant topology concepts for those subjects is better covered in analysis books.

An interesting area where topology appears is the analysis of the shape of the universe (see )
 
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  • #8
haushofer
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Where is topology useful in physics?
Just to answer this question (the rest has been covered by others here): one field is topological field theory.

E.g., an active field of research is quantum gravity in 2+1 dimensies (2 space, 1 time). However, from differential geometry you can show that in 3 dimensions the Riemann tensor and Ricci tensor have the same number of independent components and that the Riemann tensor can uniquely be expressed in terms of the metric and Ricci tensor. This is important, because the Einstein equations say that the Ricci tensor of spacetime in vacuum is zero. In 3 dimensions this means that then also the full Riemann tensor is zero! And this means on its turn that, up to topological defects, there is no gravity because spacetime is flat.

It also means that spacetime geometry (=gravity) does not depend on the metric anymore, as in normal GR, but that only the topology becomes important. You can also show that GR in 2+1 dimensions can be described by a topological field theory called "Chern-Simons theory".In these theories the dynamics is not described by the metric, but by the topology of the manifold on which you're integrating your action.

And for your comfort: I've always considered topology to be one of the hardest pieces of math I've ever encountered, because of its abstract nature. Hope this clarifies some things and good luck :)
 
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