Well, to me it is unclear what do you mean by "reducible metric".

On the other hand, dince the product of complete manifolds is complete, I am afraid that you have to ask the original manifold to be complete.

Thanks Miguel. That's a very good point about the need for the original manifold to be complete - I'm glad you brought that to my attention. If I were to specify that the original manifold was geodesically complete, would this then necessarily be the case?

What I mean by a reducible metric is as follows. There exists a coordinate system in which the metric has block-diagonal form, as in the attached.

Thinking about it further, given that we are talking about topological properties as well as local geometrical ones, I should probably have said that the metric can be reduced to this form everywhere on the manifold. That is, the manifold can be covered by charts, in each of which the metric can takes this form.

Does that clarify the question sufficiently?

Hello:

Yes, it clarifies the question, and the answer is very simple. Given a Lorentzian manifold (actually, any semi-Riemannian manifold, including the Riemannian case) and given any coordinate system, there is a basis of the tangent bundle defined on such coordinate neighbourhood such that the matrix of the metric g with respect to such basis B is of the form:

M(g,B) = Diagonal ( +1,...,+1,-1,...,-1). So, if e_i,e_j\in B, g(e_i,e_j)=\pm \delta_{ij}.

Actually, in the Riemannian case, you can use the "Gramm-Schimdt orthonormalization method", which works on Lorentzian manifolds, but slightly modified.

Summing up: all metrics are "reducible" in your sense.

The metric can be diagonalized locally, but not over a finite chart. At any given point, it is possible to identify a coordinate basis on the tangent space at this point which is pseudo-orthonormal. Indeed, it is possible for any geodesic to construct Riemann normal coordinates which have a pseudo-orthonormal basis along this entire geodesic, but the basis will not be pseudo-orthonormal away from the basis unless the manifold is flat. A curvilinear coordinate system can be used for an extended chart, which may cover a significant portion of the manifold without discontinuities or degeneracy, but it won't have an orthonormal basis for any finite neighbourhood. For example, on a two-sphere, one may use a projective coordinate system which covers a hemisphere without discontinuities or degeneracy. Alternatively, one may use spherical polar coordinates, which have a discontinuity at phi = 0, 2 pi and are degenerate in theta at the "south pole", but are perfectly valid away from these. (This is what I mean by a finite chart - it extends in each coordinate direction, but doesn't cover the whole manifold.) However, neither has an orthonormal basis away from particular points or curves.

I'm starting to think that a reducible metric over such a chart, or even over every chart in an atlas covering the manifold, is not a sufficient condition on its own for the metric to have geodesically complete factors, even if we assume the original manifold to be complete. This seems to be just a condition on each tangent space individually, and a constraint on the derivatives on the metric is necessary to place a constraint on the overall topology. If, using the notation of the attachment above, g_ab is independent of the coordinates x^X, and g_XY is independent of x^a, then the metric connection is similarly reducible. This means that parallel transport around a closed loop must rotate a vector of the first subspace in only these dimensions and similarly for the second subspace. The holonomy group is therefore a direct product group, O(m) x O (n), where a=1,...,m and X=1,...n. (Or if the subspaces are orientable, special orthogonal groups apply instead.) Then by the de Rham decomposition theorem, the manifold has geodesically complete factors.

But unless g_ab and g_XY have these constraints applied, the space does not necessarily factorize in this way. It appears that these constraints can be applied indirectly, through field equations linking the metric with its derivatives, such as the vacuum Einstein field equations with a cosmological constant. But in the absence of such field equations/constraints, factorization isn't guaranteed.

Does this sound correct, Miguel, or anyone else?

1) Given an small open coordinate neighbourhood, it is always possible to construct a smooth orthonormal frame which is defined on the whole coordinate neighbourhood.

On the sphere, by using polar coordinates, it is possible to construct a smooth orthonormal frame on the whole sphere except one point.

2) If you want something globally defined, then you need a Cartan-Hadamard manifold, i.e., a Riemannian manifold which is complete, connected, simply connected, and whose sectional curvature is less or equal to 0 everywhere. All of them are diffeomorphic to the euclidean space.

3) On every smooth manifold, there exist an open coordinate neighbourhood whose complementary set is of meassure zero. Indeed, take a new Riemannian metric on the whole manifold. Then, construct a conformal metric which is also complete. Now, take a point and use the exponential map from this point to introduce coordinates. The set of critical points of this (exponential) map is of meassure zero. (On Cartan-Hadamard manifolds, this set is empty.)

4) Now that you can construct such an open coordinate neighbourhood, on this big set, construct your smooth orthonormal frame, with your initial metric on the whole open neighbourhood.

Hi Miguel and anyone else who may be reading this.

Best wishes of the season to you.

Let me respond first to Miguel’s last post, then extend the reflections in my last post.

Regarding your last post, Miguel, I think we may be talking at cross purposes here. By metric, I mean the field whose value at each point is given by the inner product of coordinate basis vectors – or equivalently, the components of the line element in a given coordinate system. I suspect you might be thinking of the inner product of frame bases – what you could call a “frame metric”, whereas I meant a “coordinate metric”.

Let me expand on that.

An infinity of orthonormal frames can be identified for any tangent space. As I understand it, for any open set of points on a Riemannian manifold, smooth maps between such frames on their tangent spaces can always be constructed. (The parallelism this defines gives rise to the Weitzenböck connection.) However, on a curved manifold, if this open set of points covers a region of the manifold with finite extent in each coordinate direction, these frames cannot correspond to any consistent coordinate system. They are frame bases rather than coordinate bases.

This is all explained in notes I have just published on this site:

http://bit.ly/2DQstl0

and in a very illuminating post on Stack Exchange:

https://physics.stackexchange.com/questions/324056/physical-understanding-of-coordinate-vs-non-holonomic-basis

The small open coordinate neighbourhood in your point 1) needs to be infinitesimal for the orthonormal frame to be a coordinate basis on a curved manifold.

It’s true that it’s possible to construct a smooth orthonormal frame at every on the whole sphere except one point – the exclusion of that one point being the “hairy ball theorem”. However, this won’t be a basis for any consistent set of coordinates. The basis for polar coordinates is orthogonal but it is not normalised. No matter how you modify these coordinates or carry out transformations from their bases, it is not possible to find an orthonormal coordinate basis for anything more than a geodesic. (I attempted something similar this year – finding a symmetric, metric-compatible connection based on a parallelism between orthonormal bases across a finite two-dimensional region of the sphere – for weeks, if not months, before I realised such a thing was impossible. The above Stack Exchange post pointed me in the right direction, then using articles on teleparallelism I came to realise that the connection I was looking at was the Weitzenböck connection and this is not symmetric.)

Now I’d like to turn back to the considerations in my last post. I’ve managed to answer my own question and I thought it might be useful to put my further reflections on it here, for anyone who might read it later.

The key issue seems to be the difference between an isometry and a homeomorphism, and is best illustrated by thinking about tubes embedded in R^3. A cylinder is a tube of constant radius, which we’ll denote r. If it runs along the z-axis, it can be parametrised using z and an angular coordinate phi. It is isometric to R^1 x S^1. In these coordinates, it has a metric g_{ab} = diag (1, r^2). The components of the Levi-Civita connection are all zero.

Now consider a tube of varying radius, r = r (z). One can still use the same coordinates, but now the metric in these coordinates is g_{ab} = diag (1, r^2 (z)). It is therefore no longer isometric to R^1 x S^1, but it is homeomorphic to the cylinder. The Levi-Civita connection now has non-zero components. This reflects the fact that the tube has intrinsic curvature: geodesics diverge and converge along its length. This is therefore an example of a space which has reducible metric which is not (isometric to) a product of factor spaces.

Indeed, it’s one of particular interest to me, researching spontaneous compactification. The original Kaluza-Klein theory had an analogous topology, which was the product of Minkowski space and S^1. According to Coquereaux & Esposito-Farese (https://eudml.org/doc/76480), Jordan and Thiry made a modification to this which allowed explicitly for a “a scalar field which, roughly speaking, describes how the size of the 5th dimension (the radius of a small circle) changes from place to place”.

Best wishes

Tom

There is no infinity; orthonormal frames are finite in geodesically (rugby ball cf. Grigori Perelman's Poincare conjecture solution) curved space, before an observer's identity might become linear (upon 'death,' before birth perhaps; nothing registering on the EEG, adhering to conservation laws with the body quite empirically remaining/still there), then a dimensionless 'point' nowhere and everywhere (timeless; simultaneity; being; one) and back again at h, 'Big Bang,' geodesic/modified figure 8 bends -- ALL at the rate or lightspeed (unattainably/mysteriously FINITE limit; fastest in a theoretical vacuum, to which all else would therefore be relative; speed=distance/time, time=distance/speed) of the golden ratio (Ulam's spiral; da Vinci), as in Einstein's imagery of riding alongside a lightbeam.

A brief follow-up to this conversation: I've now realised the full import of the points I made in my last post above. If the radius (or radii) of curvature of the compact space is (are) due to a matter distribution, then if that matter distribution varies across the non-compact space, so does the curvature of the compact space. The Levi-Civita connection then has non-zero components. The Lorentz connections associated with these can be identified with gauge fields for orthogonal symmetries associated with the compact space. See my update https://www.researchgate.net/project/Geometry-and-group-theory-of-spontaneous-compactification/update/5e2c6e01cfe4a777d407cb63 for more on this.