# Affine Geometry/Space

1. Apr 24, 2007

### pmb_phy

Many times when I see the term Affine space used, the person using it seems to define it as a space with no origin or something akin to that. Its hard to find a definition of this term except the one that says an affine space is a space with is affinely connected where affinely connected is defined prier to this definition. MTW (page 242) use the the term affine geometry as that branch of mathmatics which adds geodesics, parallel transport and curvature (shape) to a manifold. They then go on to say that that branch of mathematics which adds a metric is called Riemannian geometry.

I never could understand why people would think that an affine space is a space without an origin since the manifold on which the space is defined is simply a collection of points, anyone of which could be defined to be the origin of the manifold.

Could those who use the term "affine space" to mean a space without an origin please give me a reference to the source in which you found it defined like this? Thank you.

Best wishes

Pete

2. Apr 24, 2007

### George Jones

Staff Emeritus
Applicable Differential Geometry by Crampin and Pirani

Tensor Geometry by Dodson and Poston (A Springer yellow and white.)

These two references give equivalent, but different-looking, definitions of affine spaces.

3. Apr 24, 2007

### robphy

A Course in Mathematics for Students of Physics - Bamberg & Sternberg
Applied Differential Geometry - Burke

4. Apr 24, 2007

### pmb_phy

Since no manifold has an implicit origin then does this imply that all manifolds are affine spaces?

Pete

5. Apr 24, 2007

### mathwonk

to me, an affine space is a flat space in which no specific point has been chosen as origin. hence a curved manifold with no chosen origin is not affine, and a flat space like R^n in which (0,...,0) is the obvious preferred origin has more structure than an affine space.

so an affine space can be obtained by forgetting the origin of a vector space, or by adding an affine structure to a manifold, but in my opinion only to certain manifolds which are in some sense flattenable.

so people starting from a vector space think of forgetting some structure, whereas people in differential geometry starting from a manifold think of adding some structure, to get an affine structure.

oh and people in algebraic geometry are perverse, and speak of k^n even with its origin, as "affine space", since they are contrasting it with projective space.

i am an amateur here on the differential geometry side of things.

6. Apr 24, 2007

### mathwonk

i think affine space in diff geom, may be a space with a certain affine group of symmetries.

a space with a group acting may be called a homogeneous space, if no origin is chosen. after an origin is chosen it becoems a group. so one has the same distinction between groups and groups without origians.

if the groupo concerned is an affine group, then ropesumabkly one has affine space.

spaces with no origin but also with no group action are then not homogeneous and certainly then not affine.

i hoope this is approximately right.

so there is a hierarchy of spaces: vector spaces, which are groups of translations. affine spaces which are spoaces on which a vector space acts faithfully, and hence are vector spaces withut an origin chosen. lie groups, which are more general groups with a topological space structure. homogeneous spaces which are manifolds on which a lie group acts.

and general manifoklds on which no group acts.

but it seems to me an "affine structure" in diff geom is a bit more general than what i described as an affine space. i never quite grasped it.

7. Apr 24, 2007

### pmb_phy

Thank you all for your helpful response. I would like to ask those who have a text to quote the text where it defines affine space. Perhaps this will help me understand this better. Thanks.

Pete

8. Apr 24, 2007

### George Jones

Staff Emeritus
Last edited by a moderator: Apr 22, 2017
9. Apr 24, 2007

### robphy

Last edited by a moderator: May 2, 2017
10. Apr 25, 2007

### matt grime

The fact that any point 'could be an origin' is precisely the answer to your own question.

R^n can be viewed as a vector space, and the maps of vector spaces send (0,..,0) to 0. R^n can also be viewed as affine n-space, and this means we no longer treat (0,..,0) as special, and we care about properties that are preserved by maps sending x to x+v for some v.

You see, it is the *maps* of the object that are important, not the elements of the object. I mean, I could put any structure I like on an abstract set of points. But that doesn't mean I should.

If we just take a collection of points, like those on a manifold, then if we make it a vector space with X as an origin, and then as a vector space with Y as an origin, the two vector spaces are not equal. There may be a map from one to the other, but that map cannot be seen as a linear endomorphism of 'the' space with X as the origin to itself because it would have to send X to X. But as affine spaces, it is the same affine space - there is an affine map sending X to Y.

Last edited: Apr 25, 2007
11. Apr 25, 2007

### pmb_phy

Thanks Rob

That kin of makes sense now. If I were able to afford only one of these books at a time then which one do you recommend I buy first?

Thanks

Pete

Last edited by a moderator: May 2, 2017
12. Apr 25, 2007

### pmb_phy

So let me get this straight: Spacetime is an affine space, Minkowski spacetime is not. Correct?

Pete

13. Apr 25, 2007

### pmb_phy

From what I'm reading right now from those references kindly provided by Rob (thanks Rob!) it appears that you're somewhat correct. I can't say you're exactly correct because I have yet to absorb all your post and the textbook quotes above.

But I'm pretty sure that I'll be picking up one of those text books.

Pete

14. Apr 25, 2007

### robphy

It really depends on just what you want to learn.

George Jones's references are good references as well.
Each reference has its own strengths.

In my experience, among these four, ...

Burke is probably the most readable... for physical intuition accompanied by a unique viewpoint of related mathematical structure... but probably not for doing calculations.

Bamberg & Sternberg is fascinating reading, developing some topics in introductory physics with rather deep (and possible obscure at first glance) mathematical formulations [like treating circuit theory with algebraic topology, optics with symplectic geometry, electromagnetism and thermodynamics with differential forms]. (Burke also treats some of these topics, with less calculation.)

Crampin and Pirani was helpful to me in understanding the relationship between the tensor notations of the mathematician and the physicist... helping me do and better understand calculations. It was also helpful to me when I was first introduced to geometrical formulations of mechanics. (In addition, I've been trying to understand some papers by Pirani et al on the interpretation of various curvature tensors and hierarchy of mathematical structures in GR.)

Dodson and Poston was helpful in understanding and visualizing structures involving tensors and differential geometry. Unlike the other books, this book focuses more on General Relativity.

It might be best to flip through these books [assuming you have access to a good bookstore or university library... or try using http://books.google.com" (as I did) to "look inside"] ... or else read the reviews on Amazon and elsewhere (like, say, AJP or other journals). Here are some non-Amazon reviews that came up on a google search [including "review" with the author names]:
http://blms.oxfordjournals.org/cgi/reprint/20/2/183-a.pdf (Crampin & Pirani)
http://www.ucolick.org/~burke/forms/bamberg.html (Burke's review of Bamberg & Sternberg)

You might find some author homepages helpful in deciding:
http://www.ucolick.org/~burke/home.html
http://www.math.harvard.edu/~shlomo/
http://www.maths.manchester.ac.uk/~kd/homepage/coursenotes.html

Last edited by a moderator: May 2, 2017
15. Apr 25, 2007

### robphy

I would say that Galilean and Minkowskian Spacetimes are affine spaces.
I would say that time in Galilean physics is an affine space.
I would say that the Euclidean plane is an affine space.

If I distinguish an event to call my origin in Galilean or Minkowskian Spacetime, then I would say that "the set of spacetime displacements from that event" would form a vector space.
I would say that the space of 4-momenta at an event in spacetime is a vector space.
(I think it is correct to regard a vector-space is a particular type of affine-space.)

Last edited: Apr 25, 2007
16. Apr 25, 2007

### matt grime

I have no idea, nor inclination to bother checking. Vector spaces have vector space endomorhisms, affine spaces have affine endomorphisms. It is the maps that matter, not the underlying set of points. R^n is both an affine space and a vector space depending on the context.

17. Apr 25, 2007

### pmb_phy

As I read Schutz's textGeometrical Methods of Mathematical Physics I read on page71
With this in mind, is Minkowski spacetime an affine space, at least according to Schutz (who does not define the term)?

Pete

18. Apr 25, 2007

### mathwonk

i am a beginner here but i have the following opinion/questions:

minkowski space is an affine 4-space but is also equipped with a metric. I do not think this metric can detract from it being an afFINE SPACE, but am not sure.

i.e. to me minkowski space should be considered together with its metric, so its transformations should be not just aFFINE OneS BUT METRIC prESERVING ONES.

19. Apr 25, 2007

### pmb_phy

A metric associated with a space does not imply the existance of a point chosen to be origin. However once you start talking about coordinate transformations then it appears to me that you've adopted a coordinate system and perhaps an origin. This part I'm not sure about. Is there a origin in spherical coordinates? No.

Pete

20. Apr 25, 2007

### MeJennifer

Hi Pete,
Sure it is an affine space.
Schutz simply means that we don't need different coordinate patches to cover everything if it is a Minkowski space, since it is all flat.