# Differential Geometry: Book on its applications?

• Geometry
• s00mb
In summary, Bruce J. West's books on fractional calculus may be useful for someone interested in applications of differential geometry.
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

mpresic3, Abhishek11235, FourEyedRaven and 1 other person
Looks very promising, thank you!

Demystifier
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
Daverz said:
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

Daverz and 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.

caz said:

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.

s00mb said:
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

Last edited:
caz said:
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.

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.

love_42 said:
contains material/approach that is not generally discussed in books on differential geometry
Such as?

Demystifier said:
Such as?
Sheaves, exact sequences, cohomology.

vanhees71 and Demystifier

## What is differential geometry?

Differential geometry is a branch of mathematics that studies the properties of curves and surfaces in multi-dimensional spaces. It uses tools from calculus, linear algebra, and topology to understand the geometric properties of these objects.

## What are the applications of differential geometry?

Differential geometry has many applications in various fields such as physics, engineering, computer graphics, and robotics. It is used to model and analyze the shape and movement of objects in these fields.

## What are some examples of differential geometry in real life?

Some examples of differential geometry in real life include the design of curved roads and bridges, the analysis of fluid flow in pipes and channels, and the study of the shape of the Earth's surface.

## What are the main concepts in differential geometry?

The main concepts in differential geometry include curves, surfaces, manifolds, tangent spaces, curvature, and geodesics. These concepts are used to describe and analyze the properties of geometric objects.

## Do I need a strong background in mathematics to understand a book on differential geometry?

Yes, a strong foundation in calculus, linear algebra, and topology is necessary to fully understand a book on differential geometry. However, there are also introductory texts available that require only basic knowledge of these subjects.

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