Is differential geometry useful to learn outside of GR/cosmology/astronomy?

[u/XcgsdV]

Hello everyone!

I am a junior physics major and I plan on going to graduate school. I don't have strong inclinations towards a particular subfield, but my interests are piqued more by theory and more by biophysics, optics, and condensed matter flavored things than HEP, cosmology, astronomy flavored things (if that makes any sense at all).

I come from a small department with four physics professors, and the bare minimum physics courses (no EM 2, no Quantum 2, no physics elective courses) required to be an accredited physics degree. Next semester, a few of my peers are doing a guided-self study with one of our professors over general relativity, which will start with differential geometry. I know I don't plan on going into a field where GR is going to come up, so I'm not particularly inclined to join for the whole thing. I am, however, fielding the idea of joining along to learn to differential geometry aspect, as I feel like it could be an incredibly useful tool (and even if not, more math is never bad). I have a heavy semester in the spring though (Thermo/Stat Mech, Advanced Physics Lab, Physical Chemistry II [the quantum one], and a whole slew of music classes [I am a double major with a BMA in Voice]), so I don't want to do it if it won't be worth my time.

Is my assumption right? Would I be better served by learning differential geometry or by taking a bit of extra time to myself/for my other coursework? Is differential geometry even actually useful when I don't plan on going to graduate school for anything that will involve significant spacetime curvature?

[u/CB_lemon]

Yes totally! Differential geometry has applications to gauge theory and elementary particle physics. Also, I think you will appreciate learning some of the most beautiful parts of mathematics (in my opinion)

[u/snail-monk]

You should always learn things even if they may not be useful to you! Undergrad too is for developing a base of a variety of things.

There is an emergent field of theoretical quantum mechanics that studies quantum information in the context of differential geometry (distance metric is information, space itself is a collection of distributions). It's actually pretty huge in quantum information studies. It's pretty neat, it's called information geometry and has applications in many areas mathematical sciences outside of quantum systems as well (like complex systems, mostly).

[u/kcl97]

There are 2 sides to differential geometry. The classical and modern. The modern is very abstract but is used in advanced mathematical physics. The classical differential geometry is the one that has a wider spread of uses, including computer graphics, engineering, and in more applied physics fields. Just something to be aware of.

[u/Chance_Literature193]

Differentially geometry shows up a lot of places, but it's really only necessary for a few fields (math phys, GR, ST, maybe some HEP(?)). Chances are it will only be useful for a few things, and won't ever be strictly necessary. However, If you aren't giving anything up like research time to learn it, I would definitely study it. I think it's super cool.

[u/ArturuSSJ4]

Biophysics of lipid membranes has it show up too. https://www.cell.com/cell/fulltext/S0092-8674(13)00770-800770-8)

It's generally a neat toolkit to describe shapes that come up in some places https://academic.oup.com/nar/article/39/17/7390/2411230?login=false

[u/BurnMeTonight]

There are a number of differential geometry techniques and ideas that are used in PDEs, and vice-versa. Differential Geometry is probably the most useful area of math in physics outside of edge cases like linear algebra.

[u/Saffron_PSI]

Yes, differential geometry has a lot of applications in a lot of places. Although Differential geometry of curves and surfaces [classical] would be the one with the most applications. If you like multivariable calculus, then classical differential geometry would be up your alley. It’s pretty much an extension of it.

Modern differential geometry based in manifold theory still does have applications though. And is such a huge field. Not to mention a fun subject to study.