Is it necessary to study Euclidean Geometry before Differential Geom.?

[Tosh5457]

Hi, I'm a Physics undergraduate, and this semester I have the option to choose between Geometry (Axiomatic Euclidean Geometry) and other disciplines. In the next year I want to be ready to study Differential Geometry, but I don't know if I need to study Euclidean Geometry first. The teacher of D. Geometry told me it would help, but I really hate studying Euclidean Geometry, so if I could avoid it the better. What do you think?

 

[micromass]

No, you really don't need to study it beforehand. Of course, you need to be comfortable with basic analytic geometry such as finding equations of lines and circles. But I assume you know that from high school.

More important is calculus and linear algebra. That is going to be used quite heavily.

 

[Tosh5457]

No, you really don't need to study it beforehand. Of course, you need to be comfortable with basic analytic geometry such as finding equations of lines and circles. But I assume you know that from high school.

More important is calculus and linear algebra. That is going to be used quite heavily.

Thank you. My teacher also told me that I'll need topology, does differential geometry use it a lot?

 

[micromass]

Thank you. My teacher also told me that I'll need topology, does differential geometry use it a lot?

Depends on the course. Some differential geometry courses require no prerequisite knowledge of topology and they introduce the necessary concepts during the course. Other courses use it extensively.

But if your teacher says you'll need topology, then you're probably going to use it a lot.

What book is your teacher going to use? Maybe you can read through his book or his syllabus in order to see if you're in over your head or not.

 

[Tosh5457]

Depends on the course. Some differential geometry courses require no prerequisite knowledge of topology and they introduce the necessary concepts during the course. Other courses use it extensively.

But if your teacher says you'll need topology, then you're probably going to use it a lot.

What book is your teacher going to use? Maybe you can read through his book or his syllabus in order to see if you're in over your head or not.

I'll check that. The teacher might change next year anyway, so I'll have to wait and see.

 

[dh363]

A lot of undergraduate differential geometry doesn't require much more than strong multivariable calc and linear algebra, and some mathematical maturity. Generally, it will be something like a "differential geometry of curves and surfaces" course taught from books like Do Carmo, O'Neill, etc. These books develop DG from a much more concrete, intuitive cases, and them abstracting them from there. They do build up some topological machinery, but never really to the full abstraction that's needed in a full course on topology. They usually just explain such concepts just enough to be applied to the course. For higher level DG, you need to be well acquainted with the language of smooth manifolds and such, which does require a bit more topological background.

The best thing to do would be to find out the book you're using, and see if you can get a syllabus from past classes. I actually found it useful to have learned DG from O'Neill before learning things like smooth manifold theory and more advanced diff geometry, because it provided motivation and examples of concepts I hadn't seen yet in full rigor.