Different types of differential geometry?

[dyn]

I am planning on taking a course in differential geometry. I have looked at the notes and they cover - differential forms , pull-backs , tangent vectors , manifolds , Stokes' theorem , tensors , metrics , Lie derivatives and groups and killing vectors. I have a book called Elementary Differential Geometry by Pressley but it contains none of the above subjects apart from tangent vectors. So my question is ; are there different types of differential geometry and what is the type called that I am looking at.

 

[Matterwave]

Everything you listed would be under the umbrella of differential geometry, with Lie groups obviously in addition involving group theory and not just differential geometry. In addition, those subjects form basically the core of differential geometry (except maybe the differential forms and the Lie groups stuff which are a little bit more advanced). If your book mentions none of those subjects...I would find it very hard to believe its a book on differential geometry! Perhaps the topics are covered but the chapters are named differently for whatever reason?

 

[dyn]

I looked in the index and none of those areas are mentioned except for tangent vectors whereas I looked in the index of Geometrical Methods by Schutz and they are all included.

 

[Matterwave]

That's very odd indeed! I don't know this book by Pressley though so I can't really make any concrete comments about it. All I can say is that differential geometry usually starts off with manifolds and their definition, proceed to tangent vectors and tangent spaces, one forms and cotangent spaces, and finally tensors, Lie derivatives and flows.

And then depending on how much time the student has, one usually then sees more advanced topics like differential forms, volume elements, integration on manifolds, Stoke's theorem, affine connections, Riemannian and Pseudo-Riemannian manifolds with metric.

 

[dyn]

Thanks. Maybe the difference is that the Pressley book appears to be more on the "pure" side while the notes and the Schutz book come from the mathematical physics side of things.

 

[Matterwave]

Can you maybe list some major topics that Pressley's book does mention?

 

[mathwonk]

differential geometry is about curvature. see if that appears in the index.

 

[dyn]

I havn't studied differential geometry before so maybe things have different names but in the index of his book there is no mention of differential forms , pull -backs , manifolds , Stokes' theorem , tensors , metrics , Lie derivatives or killing vectors.

 

[dyn]

differential geometry is about curvature. see if that appears in the index.

Yes curvature is in the index. Curvature of a catenary , curve , helix , asteroid and surface

 

[Blazejr]

I don't know this book you have either, but what comes to my mind is that it might be about stuff like surfaces and curves in higher dimensional Euclidean spaces. One can see curved surface as being drowned in three dimensional Euclidean space. That is old fashioned approach to differential geometry. If the surface alone is only thing that is really intersting to us then we want to forget about surrounding space and study properties of the surface alone. That is made precise by notion of a manifold. This approach is somewhat more abstract, but way more powerful.

 

[WWGD]

Yes curvature is in the index. Curvature of a catenary , curve , helix , asteroid and surface

But note that much of Riemannian geometry has to see with curvature in higher dimensions, i.e., dimensions 3-and-higher, where you need to bring in tensors and other tools, because there are many more directions to consider. AFAIK, the classical d.g. deals with curves and surfaces, and the more modern stuff deals with manifolds in general ; 3- and higher dimensional manifolds.

 

[dyn]

Looks like my Pressley book only deals with 3-D space so maybe doesn't need to use manifolds and differential forms etc. But my notes are concerned with mathematical physics so needs higher dimensions and thus needs manifolds etc.

 

[lavinia]

I am planning on taking a course in differential geometry. I have looked at the notes and they cover - differential forms , pull-backs , tangent vectors , manifolds , Stokes' theorem , tensors , metrics , Lie derivatives and groups and killing vectors. I have a book called Elementary Differential Geometry by Pressley but it contains none of the above subjects apart from tangent vectors. So my question is ; are there different types of differential geometry and what is the type called that I am looking at.

Classical differential geometry of surfaces is presented only using calculus in three space. The ideas of manifold,tensors,metrics are unnecessary.

Modern books do treat surfaces in 3 space using these ideas. Compare Struik's book on classical differential geometry with Singer and Thorpe's book to see the difference in approach.

 

[homeomorphic]

In my opinion, it's better to start with a book like that because it's more down to Earth and you see where all the more modern concepts come from. So, don't get hung up on the fact that you "need" higher dimensions and manifolds right off the bat. You can understand all that stuff much better if you know curves and surfaces in R^3.