coordinate based vs non-coordinate based differential geometry

Mike706

Hello Everyone,

I am just wondering what the difference in these is. Could someone please give a brief example of non-coordinate based differential geometry vs the equivalent in coordinate based, or explain the difference (whichever is easier)? Also, what advantages does one have over the other?

Thanks for your help,
Mike

Edit - Also, are there specific names for these two types of DG? Thanks again.

quasar987

Hi,

There is no difference btw coordinate-free DG and coordinate DG. They are the same thing (namely, DG!) but viewed from two different perspective. The situation is exactly analogous to coordinate free linear algebra vs coordinate linear algebra. For instance, given an abstract vector space V, the coordinate-free approach sees a linear operator T:V-->V as a map s.t. T(ax+by)=aTx+bTy. The coordinate approach results from choosing a basis v1,...,vn of V, which sets up an isomorphism with R^n via v_j -->e_j. Then a linear operator T:V-->V is one such that in coordinates, if x=(x1,...,xn), then
[tex]
(Tx)_i=\sum_{j=1}^na_{ij}x_j.
[/tex]
for some set of n² numbers a_ij.

Whereas the coordinate approach of linear algebra arises from the non-coordinate one from choosing a basis for V, likewise the coordinate approach of DG arises from the non-coordinate one from choosing a coordinate system on the manifold M.

In DG as in linear algebra, sometimes it is preferable to work in the coordinate free approach either because it conveys the ideas of what's going on more transparently, or because the computations are easier/less messy to carry out. And some other times the same holds true of the coordinate approach over the coordinate free approach.

lavinia

Geometric structures can be described globally without reference to coordinates. However they can always be expressed locally in a coordinate system. In the 18'th century and maybe earlier I am not sure, geometers described curves and surfaces by parameterizations and then deduced the geometry from the parameters. This is a coordinate approach to geometry. But later is was realized that geometries exist as global structures across an entire surface and that the surface can not necessarily be described with a single coordinate system.

I can describe examples if you like.

Mike706

Thank you for the quick replies!

quazar987:

That made a lot of sense, I think I understand it now. Could you just confirm my understanding please? I am using this book. It uses Clifford algebra instead of differential forms.

-In the beginning of the book, the basis vectors were described explicitly in terms of matrices. The main purpose this is done, it is stated, is to show concretely that we can actually use matrices to form a linearly independent basis of the required degree. So using this basis is then coordinate-based DG.

-In chapter 4 the author just said we will not consider any more explicit matrices, and also states without proof (beyond scope of the book) that it is possible to define Clifford numbers without matrices at all. So we will no longer consider matrix products, but Clifford products. This is then the coordinate-free approach, since we are not considering any specific basis anymore.

Am I correct?

Edit - Actually, I'm not so sure that was correct, now that I think about it further. Would it be possible for you to provide a simple concrete example?


Lavinia:

Thank you for the reply as well. I've heard that said before in the past and have been very curious about it. However, I only started studying DG within the last few weeks so I have not come across it yet. I would definitely be interested in an example.



Thanks again
Mike


Edit - Formatting

quasar987

I am not familiar with this version of DG, but what you wrote makes sense to me.

lavinia

I recommend Singer and Thorpe's book, Lecture Notes on Elementary Geometry and Topology. the section on Differential Geometry uses the modern approach that starts with a connection 1 form on the tangent unit circle bundle of the surface. It later shows how this coordinate free method can be translated into the classical theory of parameterized surfaces in Euclidean 3 space. This is not a graduate level book. It is for beginners.

Mike706

It made sense to me, as well. But it was wrong :P.



Lavinia I think you just answered about 3 of my questions with what you said after recommending the book. Things clicked - it was good. I will definitely check out that book as well, thanks.