Differential Geometry: Book on its applications?

[s00mb]

Hi, I'm already familiar with differential forms and differential geometry ( I used multiple books on differential geometry and I love the dover book that is written by Guggenheimer. Also used one by an Ian Thorpe), and was wondering if anyone knew a good book on it's applications. Preferably not just in the realm of relativity. I used to study a lot of pure mathematics topics and now I'm leaning towards applications and I've noticed that there is very little on the subject that I can find, which I think is a shame because it is my favorite subject (Fractional calculus is cool too, I even found some papers on fractional differential forms and geometry). Any suggestions? I'm flexible on this; it doesn't have to be a dedicated differential geometry book but I'd say if it has a few good chapters on applications that would be neat. I'm not too familiar on the subject of aerodynamics, does anyone know if that subject uses it? Thanks for your help! -James

 

[Demystifier]

The book by Schutz
https://www.amazon.com/dp/0521298873/?tag=pfamazon01-20
discusses various applications of differential geometry in the following branches of physics: thermodynamics, Hamiltonian mechanics, electromagnetism, fluids and (of course) cosmology.

 

[s00mb]

Looks very promising, thank you!

 

[Daverz]

Burke's Applied Differential Geometry

https://www.amazon.com/dp/0521269296/?tag=pfamazon01-20

 

[robphy]

Possibly useful [...these are on my to do list... someday]:

Hirani, Anil Nirmal (2003) Discrete exterior calculus. Dissertation (Ph.D.), California Institute of Technology. https://resolver.caltech.edu/CaltechETD:etd-05202003-095403

Crane, Keenan. Discrete Differential Geometry: An Applied Introduction
https://www.cs.cmu.edu/~kmcrane/Projects/DDG/
https://www.cs.cmu.edu/~kmcrane/Projects/DDG/paper.pdf

Bossavit, Alain. Computational Electromagnetism: Variational Formulations, Complementarity, Edge Elements
https://www.amazon.com/gp/product/0123885604/?tag=pfamazon01-20
related:
https://www.researchgate.net/publication/254470625_On_the_geometry_of_electromagnetism
https://www.researchgate.net/publication/200018385_Differential_Geometry_for_the_student_of_numerical_methods_in_Electromagnetism
https://www.researchgate.net/publication/242462763_Computational_electromagnetism_and_geometry_Building_a_finite-dimensional_Maxwell's_house

Burke's Applied Differential Geometry

https://www.amazon.com/dp/0521269296/?tag=pfamazon01-20

The errata for the Burke's book is at
http://www.ucolick.org/~burke/forms/errata.ps
linked from http://www.ucolick.org/~burke/class/adg.html

 

[s00mb]

Thanks for the additional references. I've started the homological algebra and have previously read some of Kranes paper I like them alot. I haven't seen that before. I had a different book on discrete differential geometry but it was very jumbled with different topics kind of piled one on top of each other with no seeming attention to its order. I find the discrete stuff very interesting. I wonder if there is a discrete analog of hyperbolic geometry or if you can construct such a thing using Krane's stuff? I'll definitely read that thank you.

 

[Frabjous]

I do not know if this is still of interest to you

Differential geometry applied to classical mechanics is sometimes known as geometric mechanics.
https://www.amazon.com/dp/0387968903/?tag=pfamazon01-20
https://www.amazon.com/dp/038798643X/?tag=pfamazon01-20
https://www.amazon.com/dp/0821844385/?tag=pfamazon01-20
https://www.amazon.com/dp/0387967907/?tag=pfamazon01-20
https://www.amazon.com/dp/0521636361/?tag=pfamazon01-20
https://www.amazon.com/dp/0199212902/?tag=pfamazon01-20
https://www.amazon.com/dp/184816775X/?tag=pfamazon01-20
https://www.amazon.com/dp/1848167784/?tag=pfamazon01-20
https://www.amazon.com/dp/3319569511/?tag=pfamazon01-20

Control Theory
https://www.amazon.com/dp/1493930168/?tag=pfamazon01-20

Solid Mechanics
https://www.amazon.com/dp/9814616036/?tag=pfamazon01-20

Differential geometry applied to information theory is known as information geometry.
https://www.amazon.com/dp/B01BDF0CQM/?tag=pfamazon01-20
https://math.ucr.edu/home/baez/information/

Bruce J. West has written several books on fractional calculus. I know that he cares about applications.

 

[s00mb]

I do not know if this is still of interest to you

Differential geometry applied to classical mechanics is sometimes known as geometric mechanics.
https://www.amazon.com/dp/0387968903/?tag=pfamazon01-20
https://www.amazon.com/dp/038798643X/?tag=pfamazon01-20
https://www.amazon.com/dp/0821844385/?tag=pfamazon01-20
https://www.amazon.com/dp/0387967907/?tag=pfamazon01-20
https://www.amazon.com/dp/0521636361/?tag=pfamazon01-20
https://www.amazon.com/dp/0199212902/?tag=pfamazon01-20
https://www.amazon.com/dp/184816775X/?tag=pfamazon01-20
https://www.amazon.com/dp/1848167784/?tag=pfamazon01-20
https://www.amazon.com/dp/3319569511/?tag=pfamazon01-20

Control Theory
https://www.amazon.com/dp/1493930168/?tag=pfamazon01-20

Solid Mechanics
https://www.amazon.com/dp/9814616036/?tag=pfamazon01-20

Differential geometry applied to information theory is known as information geometry.
https://www.amazon.com/dp/B01BDF0CQM/?tag=pfamazon01-20
https://math.ucr.edu/home/baez/information/

Bruce J. West has written several books on fractional calculus. I know that he cares about applications.

Yes I am still interested, which one would apply the most of geometric arguments in your opinion(in regard to the dynamics books)? I've seen stuff on information geometry, personally I prefer stochastic geometry over that. I am a little biased though because I have more experience with topics relating to the latter though.

 

[Frabjous]

Yes I am still interested, which one would apply the most of geometric arguments in your opinion(in regard to the dynamics books)? I've seen stuff on information geometry, personally I prefer stochastic geometry over that. I am a little biased though because I have more experience with topics relating to the latter though.

I’ve gotten interested in advanced mechanics over COVID, so you are seeing a list of interesting things that I have found. I apologize for not having read and absorbed them all so that I can give a good review

 

[s00mb]

I’ve gotten interested in advanced mechanics over COVID, so you are seeing a list of interesting things that I have found. I apologize for not having read and absorbed them all so that I can give a good review

That's no problem, the one about dynamics on manifolds seems to be the one for me to look at. I like manifold theory too, I imagine they'd apply some tensors or Riemann geometry in it.

 

[Frabjous]

These discussions might interest you
https://mathoverflow.net/questions/72160/maxwells-equations-and-differential-forms/281973

https://www.physicsforums.com/threads/covariant-macroscopic-electromagnetism.768093/#post-4837970

 

[dx]

Check out "Global Calculus" by S. Ramanan. It is not about applications, but contains material/approach that is not generally discussed in books on differential geometry.

 

[Demystifier]

contains material/approach that is not generally discussed in books on differential geometry

Such as?

 

[martinbn]

Such as?

Sheaves, exact sequences, cohomology.

 

[dx]

Such as?

Have you heard of a 'structure sheaf'?

https://en.wikipedia.org/wiki/Differentiable_manifold#Structure_sheaf

It is a really awesome book.