Should I take a course in differential geometry?

[barboza.g]

Hi guys,

I'm thinking of maybe of studying differential geometry as part of my undergraduate degree. However, it's not for physicists, it's a full on formal mathematics course specifically for mathematicians. I'm not sure whether it's a bit overkill and won't actually be useful. We don't have a course for physicists at my university; however, I do plan on taking general relativity to which includes dif. geometry at the beginning; maybe that's enough. I wanted to do something in string theory for my undergraduate thesis although I'm not sure what yet.

To actually take a class in differential geometry first you have to take a semester in "projective geometry" (I've been told it's not really projective geometry) which includes:
-Affine spaces
-Curves & regular surfaces
-Gauss maps
-Intrinsic geometry (conformal maps,geodesics,Gauss-Bonnet theorem)

Then the following semester the differential geometry course would include:
-Implicit and inverse function theorems.
-Manifolds and differential functions
-Partitions of unity, quotients and group actions
-Tangent bundles & fields
-Lie groups
-Differential forms and orientability
-de Rham cohomology
It's supposed to include Riemannian manifolds but the courses vary depending on which professor gives them.

In general relativity (which I'm doing for sure) I get a 5-week course in:
-Topological space. Differential manifold. Tangent and cotangent spaces. Tensors and p-forms. Areas and volumes. External derivative. Closed and exact forms. Poincaré, Frobenius, Stokes.
-Lie derivative. Hodge star operator. Covariant derivative. Torsion. Normal Riemann coordinates. Riemann and Ricci tensor.

Links in spanish:
-http://cms.dm.uba.ar/academico/programas/Geometria_Proyectiva and http://www.dm.uba.ar/materias/geometria_proyectiva/2012/2/
-Differential geometry syllabus and http://mate.dm.uba.ar/~pzadub/2015_1_geodif/
-http://materias.df.uba.ar/rga2015c1/programa/ and http://materias.df.uba.ar/rga2015c1/guias/

 

[micromass]

To actually take a class in differential geometry first you have to take a semester in "projective geometry" (I've been told it's not really projective geometry) which includes:
-Affine spaces
-Curves & regular surfaces
-Gauss maps
-Intrinsic geometry (conformal maps,geodesics,Gauss-Bonnet theorem)

You're right, this is not projective geometry at all. I don't understand why they name it that way. But anyway, ...

So this really depends on you. Many people approach the heavy mathematics in GR very differently. This is because many people have very different needs apparently.

It sounds like the GR course would just give you some intuitive understanding of the concepts involved. I'm sure that it won't go deep, or be very formal. Now, some people are ok with intuitive understanding. Other people find it horrible and look for more formal and deep explanations of the concepts. You should probably already know which one you are.

If you want to understand the nitty gritty details of the mathematics of GR, it seems like the two courses you mentioned are excellent. They'll teach you the logic behind the concepts. I'm sure this course is not absolutely necessary for you to take, but only you should know whether you're better off with this or not.

 

[Andy Resnick]

Hi guys,

I'm thinking of maybe of studying differential geometry as part of my undergraduate degree. However, it's not for physicists, it's a full on formal mathematics course specifically for mathematicians. I'm not sure whether it's a bit overkill and won't actually be useful. We don't have a course for physicists at my university; however, I do plan on taking general relativity to which includes dif. geometry at the beginning; maybe that's enough.
<snip>

Is there a continuum mechanics/fluid mechanics course taught by the Engineering college? I ended up taking a continuum mech. class from the Materials Science department that covered these topics.

 

[barboza.g]

If you want to understand the nitty gritty details of the mathematics of GR, it seems like the two courses you mentioned are excellent. They'll teach you the logic behind the concepts.

Excellent! I was afraid of it being too general for what I was looking for. I really am looking forward to taking the course though, I do enjoy those nitty gritty details.

Is there a continuum mechanics/fluid mechanics course taught by the Engineering college? I ended up taking a continuum mech. class from the Materials Science department that covered these topics.

I had no idea, I checked just now and we actually do have a continuum mechanics course at Engineering which covers these topics. Looks quite interesting.

You're right, this is not projective geometry at all. I don't understand why they name it that way. But anyway, ...

Yes, our department of mathematics has a problem with names sometimes. Another impressive example is a linear algebra course for biologists called, for some unfathomable reason, "Numerical Calculus".