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Die Herren Rubens und Mic
science forum beginner
Joined: 14 Jun 2006
Posts: 3
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 Posted: Wed Jun 14, 2006 8:42 pm Post subject:
Affine and projective manifolds
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It happens that one is given a differential manifold M of dimension n
and a (path dependent) linear mapping between all pairs of tangent
spaces
X : TM_x -> TM_y.
Is it not weird that this is usually called an affine connection?
Surely linear connection is the correct term? Indeed in Springer's EoM
( http://eom.springer.de/L/l059140.htm ) this is the definition.
But then their definitions of affine and projective connections (
http://eom.springer.de/a/a010950.htm and
http://eom.springer.de/p/p075180.htm ) say that M is to be taken as
base space in a bundle with fibers being affine or projective spaces of
dimension n and a connection is then an affine/projective mapping
between the fibers. Well, this is even weirder. Suddenly a massive
amount of affine and projective spaces just pop into existence. But,
why? Yes, oh why, dear Bourbakists.
Nah, let us try to motivate this a bit different. Take M as a
differential manifold having instead dimension n+1. Assume that we
would like to study curves coming from inertial motions in M (no, that
is not geodesics, I didn't say metric, did I). Then we are not really
interested in the tangent spaces TM but rather in the projectivized
tangent spaces PTM and a mapping between these:
X : PTM_x -> PTM_y,
i.e. a projective connection on M.
If it is the case (and it always is of course) that the manifold is
(n,1)-anisotropic, i.e. n dimensions are the "same" and one is
"special", then in every tangent space TM_x, there is a distinguished
affine subspace (A_n)_x of dimension n. This could also be constructed
from PTM_x by dropping the (non-physical) points at infinity. Then the
projective connection turns into a mapping
X : (A_n)_x -> (A_n)_y
which is the corresponding affine connection on M.
Finally, if we forget the special dimension by projecting
M^n+1 -> M^n
then the affine spaces become n dimensional tangent spaces
(A_n)_x = TM_x
and the affine connection on M^n+1 becomes
X : TM_x -> TM_y,
a linear connection on M^n.
Of course, we have thrown away a lot of things in arraiving at this.
Who nows, maybe there is physical insight in staying in the projective
setting. |
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Ken Pledger
science forum Guru Wannabe
Joined: 04 May 2005
Posts: 268
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| Posted: Thu Jun 15, 2006 1:33 am Post subject:
Re: Affine and projective manifolds
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In article <1150317738.935320.72530@g10g2000cwb.googlegroups.com>,
"Die Herren Rubens und Michel" <rubensmichel@yahoo.com> wrote:
| Quote: | It happens that one is given a differential manifold M of dimension n
and a (path dependent) linear mapping between all pairs of tangent
spaces
X : TM_x -> TM_y.
Is it not weird that this is usually called an affine connection?
Surely linear connection is the correct term? ....
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I'm not at all sure about this, but it could be a survival of the
very first use of "affine" by Euler, "Introductio in Analysin
Infinitorum," Volume 2, 1748, Chapter 18, "De Similitudine et Affinitate
Linearum Curvarum." He defines "curvae affines" in section 442.
If the modern use of "affine connection" on manifolds is in fact a
generalization from Euler's curves, then it actually has an older
pedigree than the other better-known modern uses of the word "affine".
But someone who knows more about manifolds may be able to correct
me.
Ken Pledger. |
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Die Herren Rubens und Mic
science forum beginner
Joined: 14 Jun 2006
Posts: 3
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| Posted: Fri Jun 16, 2006 3:55 pm Post subject:
Re: Affine and projective manifolds
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Ken Pledger wrote:
| Quote: | In article <1150317738.935320.72530@g10g2000cwb.googlegroups.com>,
"Die Herren Rubens und Michel" <rubensmichel@yahoo.com> wrote:
It happens that one is given a differential manifold M of dimension n
and a (path dependent) linear mapping between all pairs of tangent
spaces
X : TM_x -> TM_y.
Is it not weird that this is usually called an affine connection?
Surely linear connection is the correct term? ....
I'm not at all sure about this, but it could be a survival of the
very first use of "affine" by Euler, "Introductio in Analysin
Infinitorum," Volume 2, 1748, Chapter 18, "De Similitudine et Affinitate
Linearum Curvarum." He defines "curvae affines" in section 442.
If the modern use of "affine connection" on manifolds is in fact a
generalization from Euler's curves, then it actually has an older
pedigree than the other better-known modern uses of the word "affine".
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True, but I believe "affine connection" is an invention of Cartan, so
it should be possible to work out a fully coherent understanding of
this in a modern context.
As I see it connections are all about mappings between tangent spaces
and since tangent spaces are linear spaces (in essence - per
definition) the most general connection cannot be anything else but a
linear mapping. To qualify a connection as projective, affine, etc.
means to specialize the allowable mappings further (projective =
equivalence classes of linear connections, affine = linear connections
that preserve some affine subspace).
It does not make sense to define an affine connection as an affine
mapping between tangent spaces. The tangent spaces are not affine
spaces in any natural way.
It is interesting to consider the case of an invariant torus in the
cotangent bundle. On such a torus it is impossible to stand still, one
is automatically translated along it. So, here we have an affine
mapping acting. But this affine mapping is acting on the torus itself
(which is an affine space). The connection associated with the tangent
spaces of the torus is utterly dull, it is identity.
Also, the idea that homogeneous spaces are constructed from taking
quotients of Lie groups might sound good from an algebraic viewpoint,
but geometrically this is not the correct way to think about them.
Homogeneous spaces are created by moulding them from vector spaces that
happen to be around. Such as projectivizing a tangent space, picking an
affine subspace, or finding a torus in a vector bundle. |
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Lee Rudolph
science forum Guru
Joined: 28 Apr 2005
Posts: 566
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| Posted: Fri Jun 16, 2006 4:54 pm Post subject:
Re: Affine and projective manifolds
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"Die Herren Rubens und Michel" <rubensmichel@yahoo.com> writes:
....
| Quote: | As I see it connections are all about mappings between tangent spaces
and since tangent spaces are linear spaces (in essence - per
definition) the most general connection cannot be anything else but a
linear mapping. To qualify a connection as projective, affine, etc.
means to specialize the allowable mappings further (projective =
equivalence classes of linear connections, affine = linear connections
that preserve some affine subspace).
It does not make sense to define an affine connection as an affine
mapping between tangent spaces. The tangent spaces are not affine
spaces in any natural way.
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Well, there's "natural" and then there's "natural". Certainly
*every* vectorspace is an affine space in the natural way that
consists of applying to it the forgetful functor from the category
of vectorspaces to the category of affine spaces: essentially,
forget that its origin is in any way privileged. ... But I guess
I have to agree that, certainly in the context of vectorspaces that
arise as tangent spaces of manifolds, there's nothing "natural"
about forgetting the origin!
| Quote: | It is interesting to consider the case of an invariant torus in the
cotangent bundle. On such a torus it is impossible to stand still, one
is automatically translated along it.
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Please expand on this (to me) mysterious statement. (Be gentle,
I am not a physicist nor even someone who claims to understand
mechanics. I'm not even a Riemannian geometer...)
| Quote: | So, here we have an affine
mapping acting. But this affine mapping is acting on the torus itself
(which is an affine space). The connection associated with the tangent
spaces of the torus is utterly dull, it is identity.
Also, the idea that homogeneous spaces are constructed from taking
quotients of Lie groups might sound good from an algebraic viewpoint,
but geometrically this is not the correct way to think about them.
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Here I must certainly agree. (And I would go further, perhaps, and
say that the *right* things to start with are *locally homogeneous
spaces*.) It should be a (beautiful, striking, useful, but NOT
tautologous) theorem that a manifold satisfying certain geometrical
properties turns out to be construct*ible* by the said algebraic process.
| Quote: | Homogeneous spaces are created by moulding them from vector spaces that
happen to be around. Such as projectivizing a tangent space, picking an
affine subspace, or finding a torus in a vector bundle.
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And of course they arise in many other ways too.
Lee Rudolph |
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Die Herren Rubens und Mic
science forum beginner
Joined: 14 Jun 2006
Posts: 3
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| Posted: Sun Jun 18, 2006 6:25 pm Post subject:
Re: Affine and projective manifolds
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Lee Rudolph wrote:
| Quote: | "Die Herren Rubens und Michel" <rubensmichel@yahoo.com> writes:
It is interesting to consider the case of an invariant torus in the
cotangent bundle. On such a torus it is impossible to stand still, one
is automatically translated along it.
Please expand on this (to me) mysterious statement. (Be gentle,
I am not a physicist nor even someone who claims to understand
mechanics. I'm not even a Riemannian geometer...)
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Here's the gentlest possible example: Take a mass-spring system. Its
tangent bundle is the x,v-plane. The motion of the system traces out an
ellipse clockwise in this plane and this is the invariant torus. Let's
freeze time, we are then standing at some point (x,v) on this ellipse.
Now remember that v *is* a velocity, so assuming it is nonzero, as soon
as we tick the clock we *will* move in the x-direction. Also remember
that the spring force is proportional to the displacement, and assuming
it is non-zero, v will then change when we tick the clock. In
mathematical terms this is:
x' = v
v' = -(k/m) x
As a matter of fact one sees that this actually says that the there is
only one allowed vector on each tangent space to the ellipse, (x',v')
is uniquely given by (x,v). This is the ultimate non-holonomic
constraint! (One might say that since we have considering tangent
spaces once - forming the tangent bundle - there is no need to go to a
second level tangent space. In Newtonian mechanics everything happens
in the first-order tangent spaces: the (co)tangent bundle)
But there is actually a slight nuissance. Although there is one and
only one allowed velocity in each tangent space, they have different
magnitudes. So, the connection is not trivial, the tangent vector is
stretched and squeezed when we move along the ellipse. To remedy this
we rescale the ellipse to obtain a circle while keeping the enclosed
area constant (this is symplectic geometry a.k.a areal geometry in
action, and strictly speaking one has to work with the 1-form momentum
instead of velocity on the y-axis). This way we homogenize the ellipse
and arrive at a true Euclidean space: the circle S^1. The connection
becomes trivial, so we throw it away and forget all about it: We no
longer think of the circle to have tangent spaces that needs to be tied
together, instead we have SO(2) acting globally on S^1.
Of course, the natural coordinates are polar coordinates (R, phi), R is
constant and phi increases steadily. Having access to R we can measure
length along the circle - the circle is Euclidean. One might wonder if
R is always fully defined, if not the circle becomes affine.
In a higher-dimensional setting one ends up with e.g. a torus S^1 x
S^1, having radii R1 and R2. If both R1 and R2 are given this is a
Euclidean space. If neither R1 and R2 are defined, the torus is affine.
If R1 and R2 are given only up to scale we get something in between a
Euclidean and an affine space, would that be a projective torus?
Sorry, I currently have not the time to answer your other comments. Too
bad, there seems to be lots and lots of fun things arround here. |
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