# I would like suggestions regarding reading about geometry and manifolds

• Geometry
• s00mb
In summary, the conversation discusses possible next steps for someone interested in manifold theory and differential geometry. Suggestions include investigating Discrete Differential Geometry, reading books on general relativity and de Sitter and AdS spaces, and exploring applications of homological algebra, such as in topological data analysis. The usefulness and validity of topological data analysis is debated, with some viewing it as a marketing strategy and others seeing it as a legitimate application of homological algebra.
s00mb
Hi, I just finished up with Riemann Geometry not to long ago, and did something with complex geometry on kahler manifolds. In your opinion what would be a next logical step for someone to study? I am very interested in manifold theory and differential geometry in general. I'm somewhat familiar with algebraic geometry as well, I notice that get thrown in sometimes with manifold theory. Main reason I'm asking is that now when I look for books to self study they seem to go off in different directions at this level. I would like other peoples opinions on what would be interesting to self study and a good book/paper related to it. Preference towards things that may have physical applications. Thanks!

jedishrfu said:
@fresh_42 may have some better suggestions here.
As so often this is a difficult to answer question. It's a bit like asking: I have visited all locations of the British Museum, which museum should I go next? @s00mb was absolutely right as he noticed
s00mb said:
they seem to go off in different directions at this level
The only hint you gave us was
s00mb said:
Preference towards things that may have physical applications.
I'm not sure whether mathematical textbooks are within this range. E.g. homological algebra could be a next step, but it is probably quite limited in physical applications. The more detailed the mathematics gets, the further away is it from applications - at least in general.

One application which came to mind was: read a book about general relativity for the physical part, and a book about de Sitter and AdS spaces for the mathematical part. A good starting point could be our insight articles on the key words "universe", "black hole(s)", and "(general) relativity".

DDG has a lot of physical applications too.

fresh_42 said:
As so often this is a difficult to answer question. It's a bit like asking: I have visited all locations of the British Museum, which museum should I go next? @s00mb was absolutely right as he noticed

The only hint you gave us was

I'm not sure whether mathematical textbooks are within this range. E.g. homological algebra could be a next step, but it is probably quite limited in physical applications. The more detailed the mathematics gets, the further away is it from applications - at least in general.

One application which came to mind was: read a book about general relativity for the physical part, and a book about de Sitter and AdS spaces for the mathematical part. A good starting point could be our insight articles on the key words "universe", "black hole(s)", and "(general) relativity".
Homological algebrar can be used in/for Topological Data Analysis.

WWGD said:
Homological algebrar can be used in/for Topological Data Analysis.
I'm still not convinced that topological data analysis makes any sense at all. To me this is a contradiction in itself and at best a complicated way to say pattern recognition. It sounds like a marketing strategy by McKinsey et al. to sell data mining tools and consultation power under a fancy name, or to generate some master thesis in statistics. In my opinion it is "Viel Rauch um nichts" (German title of "Up in Smoke (1978)) which literally means: a lot of smoke about nothing. The one who "invented" it could probably well have been in the cast of that movie.

How is persistent homology self-contradictory? Features that persist at diffetent levels /gradations are believed to be signal and those that do not are considered noise. How is this self-contradictory?Edit: I have read papers by respected topologists like Robert Ghrist from Penn on it. Not likely someone with a track record would put out several papers on something unfounded.

It is an euphemism to call it topological. It's good old data mining in a new frame: marketing, not content! Is it something else as regressions for large data with some simplicial complexes instead of straights? And what does it have to do with physics? Or differential geometry?

fresh_42 said:
It is an euphemism to call it topological. It's good old data mining in a new frame: marketing, not content! Is it something else as regressions for large data with some simplicial complexes instead of straights? And what does it have to do with physics? Or differential geometry?
How is it a euphemism? You form an actual chain complex associated with the data and compute and interpret homology groups.Edit: True there are a lot of baseless claims in new areas but I don't see this as one of them. I am addressing your claim that homological algebra has no applications.

But I don't see what you find self-contradictory about it.

WWGD said:
But I don't see what you find self-contradictory about it.
It's off topic to discuss this here. And it is my personal opinion. My understanding on what deserves the predicate topological is likely narrower than commonly used. Some pyramids don't make a topology for me. It's far closer to algebra than it is to topology, but algebraic data analysis doesn't sound as sophisticated. And I know how the big consultancies work. It's McKinsey, not Urysohn.

vanhees71
Somehow reminds me of "quantum economics" ;-))).

Wow thanks for all the replies, I left this thread for dead after a couple days didn't expect anyone else to notice it. I am looking up the de sitter spaces stuff. I've done some basic homological algebra before but I s'pose I can get back into it. I just like stuff about manifolds in general doesn't really matter what point of view it's from whether it's algebra topology or geometry. What got me interested was things like non euclidean geometry and finding out there were different geometries. Naturally that led to wanting to learn about metrics and riemann geometry which led to manifolds. I guess what I should have posted is "I am at the level of riemann geometry and kahler manifolds, if I want to know all about manifolds, does anyone have a good reading list?". But I didn't of course hindsight being 20/20. Thanks for the suggestions though, if anyone has anymore after this posting please let me know (Also, I did finsler geometry too). Thanks everyone.

What about fibre bundles?

vanhees71
What would be a good book on fibre bundles? I'd be interested in that but I usually find it hard to wrap my head around it.

s00mb said:
What would be a good book on fibre bundles? I'd be interested in that but I usually find it hard to wrap my head around it.
If you like you can read
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/#topwhere I calculated some examples on a specific manifold, the 3-sphere. There are also references, but not all of them contain fiber bundles.

I will start there (actually already started). Seems very well explained too thank you.

## 1. What is the best book to learn about geometry and manifolds?

The best book to learn about geometry and manifolds depends on your level of knowledge and the specific topics you are interested in. Some popular options include "Introduction to Topology and Modern Analysis" by George F. Simmons, "Riemannian Geometry" by Manfredo P. do Carmo, and "Differential Geometry of Curves and Surfaces" by Manfredo P. do Carmo.

## 2. Are there any online resources for learning about geometry and manifolds?

Yes, there are many online resources available for learning about geometry and manifolds. Some popular options include online lectures and courses on platforms like Coursera and edX, as well as free textbooks and lecture notes available on websites like OpenStax and MIT OpenCourseWare.

## 3. How can I apply my knowledge of geometry and manifolds in real-world situations?

Geometry and manifolds have numerous applications in various fields such as physics, engineering, computer science, and economics. For example, manifolds are used in physics to describe the curvature of space-time, in computer graphics to create 3D models, and in economics to study complex systems.

## 4. Is it necessary to have a strong background in math to understand geometry and manifolds?

While a strong background in math can certainly be helpful, it is not necessary to understand geometry and manifolds. Some basic knowledge of algebra, calculus, and linear algebra is usually sufficient to begin learning about these topics.

## 5. Can you recommend any additional resources for further reading on geometry and manifolds?

In addition to books and online resources, there are also many research papers and articles available on geometry and manifolds. Some popular journals in this field include "Journal of Differential Geometry" and "Mathematische Annalen". Attending conferences and workshops related to these topics can also provide valuable resources and networking opportunities.

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