What Are the Practical Applications of Differential Geometry Beyond Mathematics?

[Gonzolo]

Hi, I don't question the importance of DG in mathematics and General Relativity, but I am wondering where else it is used. Are there many other practical, applied uses for it in other areas of physics, science, computer programming, engineering or other that we should know about? Like aviation, or heat transfer or networks or something like that. Is it fair to say that whoever is comfortable with DG is comfortable with tensors and that knowing tensors is generally the most practical part?

 

[pmb_phy]

Hi, I don't question the importance of DG in mathematics and General Relativity, but I am wondering where else it is used. Are there many other practical, applied uses for it in other areas of physics, science, computer programming, engineering or other that we should know about? Like aviation, or heat transfer or networks or something like that. Is it fair to say that whoever is comfortable with DG is comfortable with tensors and that knowing tensors is generally the most practical part?

Tensors are also used in classical mechanics, electrodynamics and optics.

For an example in classical mechanics please see - http://www.geocities.com/physics_world/mech/inertia_tensor.htm

Pete

 

[robphy]

Aside from the usual physics applications...

Architecture (Geodesic domes).
Elasticity Theory.
Fluid Dynamics.
Navigation.
Computer Graphics and Animation.
Medical Imaging (e.g. http://www-sop.inria.fr/epidaure/personnel/thirion/diff-geom.html and http://noodle.med.yale.edu/~hdtag/ee913a/lecture_1_small.pdf )

...indeed any system of differential equations.


Some others...

"Differential Geometry and Statistics"
http://www.ramex.com/title.asp?id=4065

"Applications of Differential Geometry to Econometrics"
http://titles.cambridge.org/catalogue.asp?ISBN=0521651166

"On the geometry of the red blood cell"
http://www.bio21.bas.bg/proceedings/Proceedings_files/vol1/PGEOM03.pdf

"Geometric Methods In The Elastic Theory Of Membranes In Liquid Crystalphases"
http://www.wspc.com.sg/books/physics/3579.html

"Differential Geometry Measures of Nonlinearity with
Applications to Ground Target Tracking"
http://www.fusion2004.foi.se/papers/IF04-1179.pdf

http://www.math.niu.edu/~rusin/known-math/index/53-XX.html
makes reference to applying DG to understanding DNA

 

[Gonzolo]

Thanks guys, exactly what I was curious about. I did cross tensors in solid state physics too (anisotropic crystals).

 

[K.J.Healey]

Thanks guys, exactly what I was curious about. I did cross tensors in solid state physics too (anisotropic crystals).

Yea, that's what I do right now. Plasticity and Stress modelling, and now more boundary interface simulations (annealing twins). Fun stuff.

 

[mathwonk]

One of my friends who got a PhD in algebraic topology and studied bundles and tensors, works for a car company modeling shapes and testing for wind resistance apparently using some of these ideas.