In 2-dimensional case the answer is no. Let the (rectangular Cartesian) coordinate axes of Ox and Oy be rotated (say anti-clockwise) through an angle A. Let the Cartesian coordinates (x, y) of a point P (on the plane xOy before rotation) assume the new (Cartesian) coordinates (X, Y). Linear relations connecting 2 systems of coordinates are given by   x = X. cos A - Y. sin A,   y =  X. sin A + Y. cos A   giving 

dx =  (dX). cos A -  (dY). sin A,   dy =  (dX). sin A + (dY). cos A.

Accordingly, the (Euclidean) distance between two infinitesimal points P (x, y)  and Q (x + dx, y + dy) measured by (application of Pythagoras theorem applicable on the Euclidean plane xOy) is  (PQ)^2  =  (dx)^2 + (dy)^2

= { (dX). cos A -  (dY). sin A}^2  + { (dX). sin A + (dY). cos A}^2

= (dx)^2 + (dY)^2.

Thus, the (Euclidean) nature of the distance remains the same even after rotation of the axes in plane xOy.

Similarly, extend this discussion in 3-dimensional case.

The two dimensional case is not applicable because an analogon to the non local condition that the rotation axes is perpendicular to the current orientation and the displacement vector does not exist. Just this non local condition causes the difficulties. Without that condition you are perfectly right and the answer to the question would be "no". But because your example is not applicable it does not make sense to extend it to three dimensions.

If you consider the three dimensional space of orientations. Every possible orientation also is a possible rotation. Therefor the rotation is a displacement or an addition in that algebraic space. But the non local condition concerning the rotational axes complicates things and causes differences to the vector addition in an Euclidian space. As an example, that kind of  "addition" is non Abelian.

The problems are much deeper as you had assumed with your example.

Agree with  Wolfgang

There is a certain non commutivity for rotations

Try rotating a block of wood.first along one axis, say x, by 90 degrees, and then along a perpendicular direction, say y axis, also by 90 degrees

Then reverse the  order. You will not get the same final position.

Therefore ideas relating to Lie algebras, between rotation operators, are indeed involved. 

This is not consistent with the idea of simple adition of vectors, which is commutative. The exception is if it is just rotations about a fixed axis. Then I believe it.

Suggest Goldstein classical mechanics for reading. Then see how to really add rotations about non paralell directions.

the concept of distance is not affected by rotations.

The space of three-dimensional rotations is indeed a three-dimensional space. It is diffeomorphic to the space of 3x3 special orthogonal matrices, SO(3). The parameters governing the rotations (the rotation angles) can take any value, but many different values can produce the same rotation. For example, a rotation about the z-axis by pi or -pi or 3 pi has the same result. The rotations written as functions of the angles therefore represent a mapping from a parameter space diffeomorphic to R^3 (Euclidean 3-dimensional space) into one diffeomorphic to SO(3), which is a curved space - indeed a compact one.

A given point in the space of rotations represents a particular rotation. It is possible to define a tangent space at any such point, the elements of which are Killing vectors. It is possible to define an inner product in this space (for example, the trace of the product of two matrices - I seem to recall this is the Cartan-Killing form). This can be extended to a metric over a given chart of the manifold.

When we apply these rotations to a given initial point in a three-dimensional (Euclidean) space, we don't actually need all three angles to determine the final point. For example, if we apply each element to a point on the z-axis, any rotation about the z-axis will have no effect. The set of final points that can be reached form a (two-)sphere. The rotation angles about the x- and y-axes uniquely specify the point on this sphere at which we end up. This means that we can use these angles to parametrise the two-sphere. Thus the two-sphere is diffeomorphic to the set of rotations about axes in the x-y plane (a subspace of the full space of rotations), which is in turn diffeomorphic to the coset space SO(3)/SO(2).

I'm not clear what you mean by a displacement vector - whether this is a vector tangent to the two-sphere which you're applying the rotations, or a vector tangent to the space of rotations themselves.

However, I think you'll find most or all of what you're exploring in the notes called "Non-linearly realised O(3) symmetries" on my profile page. This includes explicit forms for the maps between these various spaces, including the embeddings of each of the two-dimensional spaces into each of the three-dimensional ones. (In a way, I hope it doesn't resolve all your questions, as I find it helpful to think beyond what I get from someone else, particularly on such a beautiful and illuminating topic as this.)

Hi Tom,

thank you for your comprehensive answer. After a short glance to your article I see, that it is just that kind of investigation I am looking for. 

To your question about the displacement vector:

Subject of the question in this thread is a possible homeomorphism between a three dimensional vector space with positions and displacements (vector addition) and a space of orientations and rotations.

Background is finding a possible structure of the geometry of the universe. The assumption is, that a closed isotropic geometry leads to non parallel geodesics with implications for the transport of volumes. If volumes become expanded on the transport, this can explain the red shift in a static geometry. 

 May be your article helps proving (or disproving)  that assumption.

If you guys are looking at operations inside a smooth manifold I do not think

you can do much better than the gauss riemann tradition, where everything has to be discussed locally.  It relies on the assumption that over only small distances or times 

the space time is either lorenz or just plane eucledian, depending if you use relativity

or non relatiity.

If you do rotations from the outside ,globally, then nothing happens to local distances.

If you do global translations from the outside, nothing happens locally.

Again, rotations are more complicated than a simple vector space.

Goldstein does do a good job of discusing the local groups, assumed for norrmal

eucledian type space, ie. this would be the very small spaces inside a distorted space.

Hi Juan,

indeed, according to the Gauss Riemann Tradition considering everything locally  we are not able to prove a nonlinear expansion of volumes transported along diverging geodesics. The local consideration leads to zero Christoffel symbols and zero Riemannian curvature.

But even the mathematical proof of the properties of the Riemannian curvature in respect to parallel transport along closed paths is an approximation, only valid for small paths.   But does differential geometry end with this approximation; or are we able to take into account nonlinear terms as well?

With your comment you accurately meet the point "restriction to local consideration" which could be the cause that mathematicians currently exclude closed curved geometries as a possible cause for the red shift.

I believe Perelman has the idea that the space time distortion wants to lower energy and relieve itself, so the whole thing expands. Ricci flow.

If you analize how GR is built, they follow the metric (ds)(ds), which is all a local analysis. Maybe it breaks down if the wrap becomes too extreme.

Hi Wolfgang

Thanks for recommending and citing my paper. When I submitted this as a pair of papers for publication, they were rejected on the basis that "...They are well written and appear mathematically correct. They are a perfectly valid piece of scholarship, accomplished and in many ways admirable. However, they present no new results. The mathematical techniques used are standard, and the physical conclusions are well known..." I'm therefore very glad that they are proving of use to someone. If you are willing and able to endorse it for arXiv - or you know someone who is - please let me know. If you have any questions on its contents, please don't hesitate to ask.

Hi Tom,

can you answer the following question:

Is the Riemannian way to consider differential geometry in a local approximation save against the possibility that nonliear terms arbitrarily are cancelled out locally but would lead to long distance effects?

If your nonlinear approach is appropriate to show this possible gap, then the argument "..and the physical conclusions are well known.." is no longer valid.

A good example is a "secant metric" with sin(x) instead of x. (The Christoffel symbols are zero in this metric)

Hi Wolfgang

I'm not sure I can give you a perfect answer to this, but I'll do the best I can.

Consider a manifold with two different coordinate systems on it, x and x'. If the coordinates in the x' system are analytic functions of those in x system, we can by definition expand the x' coordinates in a power series in the x coordinates. By considering two neighbouring points, it's possible to find the relation between a small variation in the x coordinates and a small variation in the x' coordinates. The first order term in this relation is the Jacobian matrix.

The Riemannian approach is to look at the spaces tangent to the manifold at each point. These are by definition flat spaces. At a given point, one can define a vector tangent to the curve of increasing x^1, x^2 and so on for each coordinate in the x system. These collectively are the coordinate basis for this system. An arbitrary infinitesimal displacement can then be written as a linear sum of the basis vectors corresponding to each coordinate. It can similarly be written as a linear sum of the basis vectors corresponding to each x' coordinate. Thus we have resolved the infinitesimal displacement vector into two different coordinate bases. It is easy to show that the components of the vector in the two coordinate bases are again related by the Jacobian matrix. This is used as the template for transforming components for any vector at that point.

The Riemannian approach is therefore based on the assumption that all coordinate systems can be related analytically, but in general they are non-linear functions of each other. (A linear change of coordinates is a combination of a rotation and a translation of the coordinate system.)

I'm therefore not sure what you mean by your first question. (If the paragraph above doesn't help sufficiently, could you please explain a) what you mean by "non-linear terms" - terms of what? and b) how they could cancel out locally but lead to long distance effects?)

This just deals with the tangent space at a given point. To compare tangent spaces at different points, you need a sense of what is "parallel" to what. This is where a connection is necessary. There are infinitely many ways of defining a connection for a given manifold, but the most common is to use the Levi-Civita connection. It is the only symmetric metric-compatible connection and its components are the Christoffel symbols. (I'm happy to give you a definition of "metric-compatible" if this would be useful.) The Riemann curvature tensor, which describes the curvature of the manifold, can be written entirely in terms of the Levi-Civita connection, in such a way that if the Christoffel symbols are zero, the curvature is zero - the manifold is flat space.

I'm not familiar with a "secant metric", so I'll go on what you've put by way of explanation. The metric in spherical polar coordinates on a the surface of a sphere (i.e. for theta and phi) contains sin theta, but it doesn't have zero Christoffel symbols. As explained above, zero Christoffel symbols means that the space is flat. So I can only assume that you're talking about normal flat space but with an unusual coordinate system on it, one that is reached from Cartesian coordinates by using the sin of at least one of those coordinates. Is this what you mean, and if so, what are you saying about it?

It is also worth being aware that the metric *at a given point* tells you nothing about the global shape of the manifold. However, the coordinate system that the metric corresponds to may be valid over a large region of the manifold. If the manifold has the same curvature properties over the whole manifold that it does over the region, this will determine the global shape of the manifold. I haven't managed to find much by way of mathematical theory relating local geometry to global topology, but there are some odds and ends. For example, if the Ricci tensor is proportional to the metric, the manifold is a "surface of constant curvature". Also, the concept of holonomy (see the Wikipedia article) can be useful, especially if you can apply the de Rham decomposition theorem. (Sorry if I've gone a little off-topic here and it doesn't sound familiar - it's quite possible that you wouldn't need to make use of any of this theory. I'm using it at the moment, so it's fresh in my mind!)

In summary, the non-linear approach I used is entirely in keeping with Riemannian geometry and all of the manifolds I looked at are Riemannian manifolds. (This is why I'm able to define inner products all the way through.) It's based on theory established in the 1960s to handle non-linear sigma models and spontaneous symmetry breaking, developed to bring out the group theory aspects. My paper just shows how to use these methods in an extremely simple case that is easy to visualise.

Hi Tom,

thank you for your comprehensive answer. The background of my mathematical question is a cosmological question:

https://www.researchgate.net/post/Has_anybody_analysed_a_rotational_geometry_with_a_secant_metric_as_a_model_for_the_geometry_of_the_universe

The possible physical consequence of a nonlinear geometry is a red shift, because on curved long distances the light needs more time as on a straight path. The secant metric describes the length difference between the path on a circular arc and along the secant. The secant metric leads to zero Christoffel symbols and therefore to zero curvature. One possibility is that “this zero” is the result of an approximation and a later extension/integration of this approximation to long distances cannot reproduce what has been cancelled in the approximation. Another possibility is, that the connection to the fourth dimension (time) concerning the Minkowsky metric must be reconsidered to get a long distance effect. My hope is that your non-linear approach allows some progress in these questions.

Hi Ari,

In very short terms:

With the Minkowski metric written as ds2=dt2-(dx2+dy2+dz2) we get a C4 space definition. Expressing a four dimensional position P in that C4 Minkowski space using exponentiated quaternions P=exp(t,ix,jy,kz) just provides the analogy between displacement and rotation.

With an appropriate calibration this space structure is a candidate for a (closed) large scale topology of our universe. Parallel transport properties along geodesics in that space are subject of physical interest.

But I don't see an application of mapping that C4 to R2.