# Differential geometry lecture notes

1. Dec 2, 2003

### franznietzsche

i've been trying to find a good set of lecture notes for independent study on the subject. I wen to the one in the thread on differential geometry and tensor calculus at people.hofstra.com, but it went offline while i was viewing it and i have had no luck reaccessing it.

2. Dec 2, 2003

### franznietzsche

it went back up online again, so i don't need help with that, but can someone please explain the use of the superscript notation for the local coordinates of the manifolds? It looks like it means x to the first power, x to the second power, etc. but that doesn't seem to make sense (because it is never referred to as a power). So is it a power function or is it simply like a subscript, excpet a superscript instead?

3. Dec 2, 2003

### lethe

It is not an exponent. it is just a label, like a subscript, but its a superscript instead. the reason for this is to allow the Einstein summation notation, which is quite nice, but may take some getting used to.

4. Dec 2, 2003

### Ambitwistor

It's not an exponent (power), it's just an index, like a subscript index. I haven't seen this nomenclature before, but they appear to be using subscripts to denote extrinsic coordinates in the embedding space, and superscripts to denote intrinsic coordinates in the surface.

5. Dec 2, 2003

### Loren Booda

A simple introduction to the Einstein notation and its basic applications to general relativity can be found in Introduction to Cosmology by Jayant V. Narlikar.

6. Dec 2, 2003

### franznietzsche

ok thnax that helps. This is my first experience with anything beyond fourier series and differential equations so that usage of the superscript was entirely foreign to me. Thanx for the help.

7. Jan 21, 2004

### rdt2

superscripts

Since superscripts are used to denote components of vectors, a notation I've seen (and since adopted) is to show powers in parentheses, thus:

x2 is the component of a vector x in the 2-direction

x(2) is x*x.

Cheers,

Ron.

8. Jan 24, 2004

### Arcon

See
http://www.math.odu.edu/~jhh/counter2.html
http://xxx.lanl.gov/abs/gr-qc/9712019
http://arcturus.mit.edu/8.962/notes.html
http://www.geocities.com/physics_world/ma/intro_tensor.htm

The superscripts are placed as they are in order to use the summation convention and to distinguish between two different types of objects.

A unit vector is written as ek. If the components of a vector A are Ak then

A = Ak ek

where the summation convention is used:
[qoute]
Summation Convention: If an index appears twice in a term, once as a superscript and once as a subscript, then summation is implied over the range that the indices are allowed to take on.
[/quote]

Last edited by a moderator: Apr 20, 2017
9. Jan 29, 2004

### Steven S

I recommend Bishop and Goldber'g "Tensor Analysis on Manifolds". It is really cheap since it is a Dover book and covers all the basics.

10. Apr 25, 2004

### Ebolamonk3y

What are the pre-reqs and such for Tensor analysis?

What other materials do you guys recommend for introduction to tensor analysis?

BTW, is there such a volume purchasing for Dover books or Schaums outlines? Like, can you purchase a ton of them for a discount price? Thanks in advance..

11. Apr 26, 2004

### franznietzsche

I recently purchased several from Amazon.com which had the best prices i could find (several dollars less than any other retailer).

I haven't started Tensor Analysis on manifolds (the Goldberg one), but Lovelock and Rund's Tensors, Differential Forms, and variational Principles is a very good book. The two in combination give a very nice treatment of the subject.

As for prereqs, reading the books you just need to know calculus, partial differentiation (which is just a very easy extension of one variable differentiation) and an understanding of vector analysis will help, but is not necesary as that is covered in the text.

12. Apr 26, 2004

### Ebolamonk3y

:) alright. Thanks.

13. Apr 26, 2004

### Ebolamonk3y

btw... which ones did you get from Amazon.com?

14. Apr 27, 2004

### Stevo

I found Tensor Geometry by Dodson & Poston to be a good book.

15. Apr 27, 2004

### franznietzsche

Well pretty much everything is cheaper there, at least if you're looking for physics/mathematics texts. The ones i got were Tensors, Differential Forms and Variational Principles by Lovelock and Rund and i also got Tensor Analysis on manifolds the authors of which i don't feel like looking up midnight, though i think itd Goldberg and someone else...

16. Apr 28, 2004

### H-bar None

Tensors, Differential Forms and Variational Principles by Lovelock and Rund is an excellent book. I've spent a lot of time reading books on tensors lately the light bulb lit up on this one.

Shaum's has a book on Vector Analysis and Tensors that is pretty good.

17. Aug 11, 2004

### mathwonk

Thanks for the posts and references. I want to issue a warning though on some of these. the free notes available at

http://www.math.odu.edu/~jhh/counter2.html

for example, seem devoted primarily to the old fashioned, index - dominated non conceptual approach, ignoring the meanings of the objects being discussed, at least in the first chapter I read, introducing tensors. maybe it gets more conceptual later.

This "indices only" approach however has been out of date for over 40 years in mathematical circles, and it was my impression also, that even in the book on relativity by the physicists J.A. Wheeler and Kip Thorne, written in the 70's, the modern viewpoint was used.

The site at wikipedia suggested by someone in the thread What is a tensor?, has both approaches, and tries to bridge the gap, while pointing out that the conceptual approach is pretty much the standard in advanced work nowadays.

Bear in mind that I am a mathematician, and not a physicist, but you might check out which language the modern physicists are using more today and try to learn that.

One remark is that the conceptual does allow you to understand the indexed aproach eventually, since it is just the coordinate representation of the conceptual version, but the index only approach seems to make it very hard to understand the conceptual approach, because no abstract concepts have been introduced.

I.e. the conceptual approach does contain the other, but not vice versa.

Last edited: Aug 11, 2004
18. Aug 11, 2004

### mathwonk

PS: The second reference in post 8 by Arcon, to notes of Sean Carroll of the Institute for Theoretical Physics at Santa Barbara, does explain both approaches.

http://xxx.lanl.gov/abs/gr-qc/9712019

Last edited: Aug 11, 2004
19. Jul 23, 2006