your question is interesting in that it sheds light on the difference between the local and the global point of view. In riemannian geometry, an isometric embedding is an embedding which is compatible with the metric tensor, and that's all.

 Dear Jacques,

If any conception is inconsistent, the inconsistency appears in different points of the conception. Do you know any other appearances of the Riemannian geometry inconsistency?

In the Riemannian geometry the distance between any two points is defined as an integral along the shortest curve, connecting the two points. But in the plane Lh with a hole we have another shortest curve. Then the plane Lh really will not be imbedded in L isometrically. The distance between the points of course change if there is an obstacle.

Dear Yuriy,

You are quite right. Putting my question, I kept in mind namely this circumstance. However, it looks rather strange. Making a hole, one changes geometry on the plane Lh. Is it an evidence of the Riemannian geometry inconsistency? Maybe, there is another explanation of this phenomenon.

It seems to me that is not evidence of the Riemannian geometry inconsistency. A hole change topological property of the plane.

Dear Yuriy,

Do you think, that topology is something external with respect to Riemannian geometry?

Hi Yuri,

We begin with a differentiable manifold, which is essentially a topological structure. If the topology is sufficiently nice (Hausdorff and paracompact for example) then we may endow it with a Riemannian metric inherited by each of the coordinate charts, thus giving the differentiable manifold a Riemannian structure. In this sense, the topological space is external to the Riemannian aspect. Therefore one is not free to change the topology without also changing everything that is inherited from that, including the Riemannian structure.

Cheers,

Glen

Dear Glen,

There are different ways of the Riemannian geometry introduction. The most simple is the following method. The proper Euclidean geometry GE is presented in terms and only in terms of the world function \sigma, which is connected with the metric \rho by means of relation \sigma =1/2\rho2 . Such a presentation is possible, when GE is presented in terms and only in terms of Euclidean \sigmaE. Even dimension of GE can be presented in terms of \sigmaE. Replacing \sigmaE by \sigmaR in all relations and definitions of GE one obtains the Riemannian geometry GR.

However, usually one uses infinitesimal world function \sigma(x+dx,x)=gikdxidxk. In other words, one replaces the Euclidean metric tensor gEik  by the Riemannian metric tensor gRik. In this case one needs topology or something like a topology  to agglutinate infinitesimal pieces of the Riemannian geometry. As a result one obtains the multivariant Riemannian geometry, where there are many g-vectors AB1,AB2,…at the point A, which are equal (equivalent) to some vector CD at the point C, whereas g-vectors AB1,AB2,…at the point A are not equivalent between themselves.

Gometrical vector (g-vector) AB is (by definition) the ordered set of two points A and B. Equivalence of two g-vectors AB and CD is defined by two coordinateless  equations

(AB.CD) = |AB| |CD| and |AB| = |CD|.           (*)

 where the scalar product (AB.CD) is defined via \sigma in the form 

(AB.CD) =\sigma(A,D) +\sigma(B,C)  - \sigma(A,C) -\sigma(B,D)

|AB|2 =(AB.AB)=2\sigma(A,B)

If points A,B.C are fixed, then equation (*) has in general many solutions for the point D, because there are only two equations (*), but the point D have 4 coordinates. Thus, the multivariance is a natural property of any (space-time) geometry. Only GE is single-variant .In other words, GE is a degenerate geometry. But one considers the degenerate geometry as a normal geometry. One demands usually, that the Riemannian geometry GR be single-variant. One introduces so called parallel transport, which is absent in GE. Nobody proved compatibility of parallel transport with axioms of the Riemannian geometry, although the parallel transport has been introduced to make the Riemannian geometry to be axiomatizable.

Multivariant geometry is nonaxiomatizable, because in the multivariant geometry the equivalence relation is intransitive. But in any axiomatizable conception the equivalence relation must be transitive.

Note. that the axiomatizability of the Riemannian geometry is not necessary, because it may be constructed as a deformation of GE, but not as a result of deduction from axioms. And there are no necessity to introduce the parallel transport. On one hand, the parallel transport cannot be obtained as a result of deformation of GE, because it is absent in GE. On the other hand, there is a doubt, that introduction of  parallel transport admits one to deduce GR from axioms.

Hi Yuri,

Your perspective is an interesting one!

At the first step, when you introduce the Riemannian metric on Euclidean space, topology enters the picture. If the introduction is global, then the topology is that of an $n$-plane. Otherwise, if the introduction is made locally, and satisfies standard change of coordinates rules and so on, then the topology is fixed by these rules.

Best,

Glen

Dear Glen,

It is not so simple with topology. In reality the topology influences on the local geometry. I show this in the simple example. Let us consider two-dimensional Euclidean plane. Let us form a cylinder, identifying the points (x - L,y) with points (x + L,y). Let us consider the points

P = (-L + e, 0),   R = (L - e, 0), and Q = (0,0). Let us now consider two vectors PQ and QR Coordinates of these vectors are PQ = (L – e,0),  QR = (L –e, 0). Vectors PQ = QR, if we use the rule, that vectors are equal, if their coordinates are equal. However,  the result will be another, if we use the rule

(PQeqvQR) if  (PQ.QR) = |PQ| |QR| and |PQ| = |QR|

The scalar product (PQ.QR) is calculated according to coordinateless rule

(PQ.QR) = \sigma(P,R) +\sigma(Q,Q)  -\sigma(P,Q)  -\sigma(Q,R)

=\sigma(P,R) - \sigma(P,Q)  - \sigma(Q,R)

Let e be infinitesimal quantity. The quantity \sigma(Q,R) is different in the case of the plane and in the case of  cylinder. In the case of  plane \sigma(Q,R) = (2L-2e)2. In the case of cylinder, when the points P= (-L + e, 0),   R = (L - e, 0) are very close, \sigma(Q,R) = (2e)2.

Thus, in the case of the cylinder the vectors PQ and QR  are not equal.

As far as coordinateless definition of equality is more reasonable, than coordinate definition, influence of topology appears to be rather unexpected.

In particular, it means that the Riemannian geometry is multivariant, and introduction of so called parallel transport, which suppresses multivariance is an  inadequate operation.

There is no inconsistency in your example: the distances between the same points in the plane and in the plane with a hole are not the same. A reason for this, as pointed out by others above, is that the distance between points in a Riemannian manifold is not a local concept. Your example shows that the distance associated to the induced Riemannian metric on a submanifold does not coincide with the induced distance from the ambient metric space structure. I am using the standard term 'Riemanian metric' to refer to a smooth, symmetric, positive definite covariant tensor of  type (2,0). And as you observed, on geodesically convex submanifolds, both distances agree.

The reason behind this apparently strange behavior of distances is that although the notion of distance is global (infimum of lengths of curves joining two given points), preserving the Riemannian metrics is a purely local property (an isometric embedding is a topological embedding that preserves the Riemannian metrics, a condition that can be expressed in terms of the differential of the map at every point).

Let me finish by pointing out a remarkable result by Meers-Steenrod (1939), that asserts that every isometry of metric spaces (distance-preserving bijection) between Riemannian manifolds is in fact, a smooth isometry of Riemannian manifolds. The converse is clearly true. 

Dear Joaquin,

I am understanding the reason, why the plane with a hole cannot be isometrically embedded to the plane without a hole. The point is not in this fact. In my opinion, if the distance between any two points of the pointset  \Omega is given, the geometry on \Omega is given. This definition of the geometry does not depend on topology, because the topology is taken into account, when the distance is determined. My question has been put, because the conventional description of the Riemannian geomentry is inconsistent. (See example in my previous answer to Glen Wheeler.)

Reimannian geometry (RG) is a math theory, which include Euclidean and other axiomatic/synthetic geometries formulated in the language of differential calculus. RG is not an axiomatic system and hence it is not very correct to apply to RG the term "inconsistency" in the sense as this concept is rigorously formalized in math. logic.

The notion of distance between two points is not defined for a generic Riemannian manifold. For instance, such is Euclidean plane with one point removed. Moreover, the notion of distance is irrelevant in RG.  Y. Rylov find "inconsistency" of RG in the fact that Riemannian manifolds do not behave as metric spaces (MS). But this only means that RG and MS are just different non equivalent generalisations of Euclidean geometry. So, the quesation is, which of them is adequate for physics. Up to now no arguments in favour of MS are known, while with RG physics  feels at home. For instance, classical mechanics in Hamiltonian setting as a pure RG.

Finally, the Levi-Civita connection, which is respopnsible for parallel transport, is an  immanent property of the metric tensor as, say, the curvature tensor does, etc. In other words, parallel transport is not something introduced artificially or by hands but a mechanism naturally defined by the matric tensor. So, I do not see here any "suppresion" or "inadequacy".

Frankly, I do not understand what is the motivation for this discussion. If RG is "inadequate" for somthing in physics or math this point should be indicated.RG has so deep and brilliant aceivements and fits so well all "experimental data coming from physics and math that put it in question would be even worther than putting in question the conservation energy law, which, by the way, is rigorously formulated properly in terms of differential geometry.

Math is the problem.  In all natural systems there are limits.  Fibonacci knew this when he talked about the natural growth of things but he realized that the system to calculate the next number would not have those natural limits.  A numbering system does not fail.  It continues on forever unless we impose limits.  Math is made by human beings but nature is not.  

My point here is that if we as scientist assume that math is infallible and has the correct answer all the time then we are not looking at nature.   Just like the point at which a sea shell that can not get as big as the sea it is in points out the limit to nature.  When we stop fighting over the numbers and start looking at the limits other possibilities will show up and we may get lucky and find the correct answer.

Math is not the answer it is the problem.  Newton understood this and created a math to help him solve the problem of calculating something that would take years to work on in just a few minutes.  Einstein worked with a Professor of math to come up with the equations of general relativity using a form of non Euclidean geometry that took 8 years to refine.  Throughout history when our math gets in the way of understanding we come up with new forms of math to solve the problems.  Today is no different, math is telling us things that can not or should not be true and we need to move beyond the old math and on to a new system that better approximates nature or the real world.   

Dear prof. Rylov

 in your example the embedding of Lh in L is an isometry of Riemanian Manifolds but not an isometry of Metric Spaces. A Riemannian manifold is also a metric space. However, isometric embeddings of Riemannian manifolds are not isometric embeddings of metric spaces: the distance may decrease. This phenomenon is very usual, and I do not think lead to any inconsistency. Only the definitions of isometry in RG and in Metric Spaces are different. 

Dear A.M.Vinogradov,

At first about motivation of my unusual question. First, in my opinion, the space-time geometry (it the Riemannian geometry now) must admit a coordinateless formulation. In contemporary textbooks introduction of Riemannian geometry GR begins as follows: “ Let us consider manifold of dimension n with the coordinate system on it…” Second, the Riemannian geometry is obtained as a generalization of the proper Euclidean geometry GE. It is true, that there are different methods of GE generalization. The most effective method of generalization is obtained, when GE is presented as a monistic conception, when there is only one fundamental quantity: world function \sigma = ½ \rho2, where \rho(P,Q) is distance between any two points P,Q belonging to pointset \Omega. All geometrical quantities and geometrical objects of GE are expressed via \sigmaE and only via \sigmaE. Replacing \sigmaE by world function \sigma of a generalized geometry G, one obtains all relations and geometrical objects of G. The Riemannian geometry GR is obtained by the same method, but one uses infinitesimal world function \sigma(x+dx,x)=gikdxidxk. It easy to see, that is equivalent to replacement of metric tensor gEik by gik. For usage of metric tensor gik instead of world function \sigma one needs: (1) dimension, (2) coordinate system and (3) topology. These quantities remain constant at the geometry deformation by means of the gik replacement. As a result the set of Riemannian geometries is not so powerful as the class of all possible generalized geometries. (I refer to these generalized geometries as physical geometries). This circumstance is important in the general relativity, where the space-time geometry depends on the matter distribution. One need to use all possible space-time geometries. In reality, one uses only Riemannian geometries, supposing that the Riemannian geometry is the most general type of the space-time geometry. In the extended general relativity, based on physical geometries, black holes are absent, because at the star collapse the induced antigravitation arises, which prevents from formation of the event horizon

 Induced antigravitation in the extended general relativity . Gravitation and Cosmology, 2012, Vol. 18, No. 2, pp. 107–112,( 2012).  

Third, the Riemannian geometry is a continuous geometry, whereas the real space-time geometry is discrete. The discrete geometry is such a geometry, where there is a minimal length \lambda. The discrete geometry is restricted by the condition

|\rho(P,Q)| does not belong to (0,\lambda) for any P,Q belonging to \Omega     (*)

 Usually one considers (*) as a constraint on \Omega. As a result one obtains a geometry on a lattice, which is not isotropic and uniform. In reality the condition (*) is a constraint  on \rho, whereas \Omega is fixed. It is possible, that \Omega is a manifold, where the geometry of Minkowski is given. The discrete geometry Gd may be isotropic and uniform. The space-time geometry Gd is multivariant and a free motion of elementary particles in Gd is stochastic. Statistical description of this stochastic motion leads to quantum mechanics (without quantum principles).  Discrete space-time geometry and skeleton conception of particle dynamics  International Journal of Theoretical Physics. Volume 51, Issue 6 (2012), Page 1847-1865.

The physical geometry is multivariant in general. It means that there are many vectors SR1,SR2,…, which are equal to vector PQ, but vectors SR1,SR2,…are not equal between themselves.

 The equality of two vectors PQ and SR is defined by the relation

(PQeqvSR) if  (PQ.SR) = |PQ| |SR| and |PQ| = |SR|           (**)

The scalar product (PQ.SR) is calculated according to coordinateless rule

(PQ.SR) = \sigma(P,R) +\sigma(Q,S)  -\sigma(P,S)  -\sigma(Q,R)

In general case two equations (**) have many solution for the point R at fixed points P,Q,S, because the point R is described by four coordinates. However, there are degenerate physical geometries, which are single-variant. For instance the proper Euclidean geometry is single-variant with respect to all vectors. The geometry of Minkowsky GM is single-variant with respect to timelike vectors and GM is multivariant with respect to spacelike vectors. It is a reason, why one considers, that tachyons do not exist. If oneconsiders GM as a physical geometry, then a single tachyon cannot be detected, because its world line wobbles with infinite amplitude. However, the tachyon gas can be detected by its gravitational field. The tachyon gas forms so called dark matter Dynamic equations for tachyon gas, Int. J. Theor. Phys. 52, 133( 10), 3683- 3695, (2013). The Riemannian geometry GR is single-variant with respect to vectors PQ and PR, having the common origin, but it is multivariant  with respect to vectors, having different origins. One tries to suppress this circumstance, using the parallel transport.  

In general, the contemporary mathematicians do not accept the multivariant geometries and the concept of multivariance in geometry. Why?  I should like to know the reason of such a position of mathematicians. As yet I think that it is a result of the school education, where only single-variant Euclidean geometry has been studied. I qualify this phenomenon as the Galiley phenomenon, when researchers of that time did not accept the concept of inertia. Both concepts multivariance and inertia are fundamental concepts, which cannot be reduced to other more simple concepts.

I put my strange questions, trying to understand, what is clear for researchers. I am keeping in mind that introduction of the concept inertia lasted many years.

Article Discrete Space-Time Geometry and Skeleton Conception of Part...

Article Dynamic Equations for Tachyon Gas

Article Induced antigravity in extended general relativity

In the schools only Euclidean geometry is studied. But in institutions non-Euclidean geometries are included in the program.

Dear Y. A. Rylov,

The coordinate free definitions of (pseudo)-Riemannian geometry is well known. This is a manifold supplied with a non-degenerate covariant symmetric 2-tensor. Let me skip coordinate free definition of tensors. Probably, you consulted textbooks written for physisists only. 

Also, as I draw up from your arguments that your question, in fact, is :

"whether RG is adequate to describe some situations in physics".

In other words, your question is about inadequacy of RG but not about its consistency. Let me also to stress on the word "some" because there are many situations in physics where RG works perfect. Hence if you find that RG is not adequate for something, then it is your task to develop a new, not yet known  geometry, which is adequate to the situation you are interested to describe in a rigorous mathematical manner. Actually, there were developped various geometries in order to cover various needs in physics. For instance, mind symplectic or Poisson geometries.

Unfortunately, I have no space-time possibilities to comment on all your assertions. I have chosen only a general one. Namely, while you are certain that

"... the Riemannian geometry is a continuous geometry, whereas the real space-time geometry is discrete",

I find this to be just one from the Brownian motions of ideas around relativities and QFT, which has neither some experimental evidence, nor a due mathematical realization. Also I would like to remark that the notion of a (at least. velocity) vector is inconsistent with discretness of what you call "the real space-time".

At this point I must appologise for leaving the discussion.

The Riemannian geometry should be placed in an axiomatic set theory, for instance in ZF. Since nobody has ever proved the consistency of ZF, how can one prove the consistency of the Riemannian geometry? I think that all axioms of ZF^{-} are in use in the Riemannian geometry, although it seems to me that  they are used unconsciously by most researchers.  I believe that it is impossible to prove its consistency,  as well as it is impossible to prove its  inconsistency.  Anyway, it would be good to analyse  applications of axioms of ZF to  the Riemannian geometry  and answer the question: which axiom of ZF is for what in the Remannian geometry?

Dear A. M. Vinogradov ,

It is very pity, that you cannot continue the discussion. However, participation of professional mathematicians is very valuable for me. Of course, the best method of discussion is report at a session of a seminar, but this way is shut for me. I tried to submit my report on multivariant geometry to a seminar of  mekh-mat of Moscow Lomonosov University. Looking through the text of my report, the secretary of the seminar  said: “How strange geometry! Only definitions! There are no theorems! Participants of our seminar are not interested in such geometries.” In other case I have discussed my possible report with the deputy of head of one of  chair on geometry and topology in the same University. Our discussion lasted about half a hour. My report has not been accepted, and I did not know why. I can only guess about reasons.  In such a situation I appreciate any discussion with a professional mathematician, dealing with geometry. 

As far as I understand mathematicians, they consider themselves to be responsible for quality of geometric formalism, which must be a logical conception, but they do not consider themselves to be responsible for adequacy of its application to description of physical phenomena. There are a lot of different geometries, and a physicist must  choose what of these geometries is adequate in the considered case. The approach of a physicist is another. He is interested, that the space-time geometry describes correctly the physical phenomena. Whether the geometry is a logical construction, this fact is not interested for physicist.

In your post you go around the concept of multivariance, which is a key concept of the physical geometry. Objections of mathematicians against the multivariance in the space-time geometry will be very interesting for me. Maybe, a tale on my path to the physical geometry will be useful for understanding of the concept of multivariance. It can be found in “Байка о том, как я модернизировал описание римановой геометрии, и что из этого получилось”

http://gasdyn-ipm.ipmnet.ru/~rylov/tale1r.htm . Unfortunatly, this tale is written in Russian.

I will be very obligated to you, if you find a time for looking through this tale and tell me about impression, which this tale makes.

If the hole h you mean is only a point, say that you speak about M = R^2 \ { (0,0) }, one can define distance (A, B) = infimum ( length of differential paths between A and B). With this notion of distance M is isometrically embedded in R^2, but in this case there are pairs of points which are not linked by a geodesic. Alternatively you can diffeomorphically transport the metric from a non-degenerated (half-)cone without vertex (which is embedded in R^3, say). In this situation you have geodetics between any pair of points, but the space is no more isometric with the corresponding open set in R^2, and the geodesics are no more straight lines. Evidently, one cannot have both in the same time!

 I believe that most people, among them, most scientists do not understand well enough what they talk about. This is partly why they create not fully adequate descriptions of objects they try to describe. I am interested in geometries and in physics as well; however, when sets, ordered pairs and numbers are used in geometries, I am sure that set-theoretic axioms should be investigated first. Unfortunately, many researchers ignore axiomatic set theories and do not seem to want to think about them. Only a minority of researchers can  think  more or less correctly about  axiomatic set theories that are necessary for adequate descriptions.

Distances in Riemannian geometry are not defined in terms of geodesics! They are defined in terms of a scalar product on the vector space of directions at each point. That is, distances are defined infinitesimally in terms of a quadratic form gijdxidxj. This then gives us, by integration, a definition of the length of a curve connecting two points A and B. In the case of a Riemannian manifold with holes, the shortest curve between two points will not necessarily be a “geodesic”! That is, it will not necessarily satisfy the “geodesic equation”.

Therefore, if we wish to think about Riemannian manifolds with holes, we need to be aware that some of the things we have learned about Riemannian manifolds without holes (ie., the "usual" Riemannian geometry that we've all learned) will not apply. But that doesn’t mean that “Riemannian geometry is not consistent”. The geometry of Riemannian manifolds without holes is consistent. The geometry of Riemannian manifolds with holes is also consistent.

However, you have raised an interesting point: in Riemannian manifolds with boundaries (ie., “holes”) the concept of “geodesics” becomes a bit tricky...

Differential geometers have the notion of a complete Riemannian manifold, which means that all geodesics can be continued for all parameters.   Euclidean space with a point removed is not complete, and this is what mathematicians usually say rather than saying inconsistent.

Let me recall that a theory T is called consistent if there does not exist a statement p such that both p and the negation of p are provable in T. I do not think that in this sense one can prove that the Riemannian geometry is inconsistent. This does not mean that one can prove that the Riemannian geometry is consistent.  

It seems to me that the answer of miss Eliza Wajch is the best.

Dear Eric,

My question is not quite correct. The qualification “consistent”  or “inconsistent” can be applied only to a logical construction, when all statements of the conception are deduced logically from axiomatics (several fundamental basic statements). Strictly speaking, the Riemannian geometry is not a logical construction. The proper Riemannian geometry GR can  be considered as a surface  S in the proper Euclidean geometry GEN, where N is dimension of  GEN. N is supposed to be large enough, in order GR can be embedded. in GEN. The difficult question is equivalence of  two vectors AB  and PQ at two different points A and P. Let for simplicity vectors AB and PQ be infinitesimal. Let PA be an Euclidean plane tangent to S at the point A, and PP be an Euclidean plane tangent to S at the point P. Then AB belongs to PA and PQ belongs to PP. Let AR and PF be to equal vectors in GEN. AR = PF

 Vectors AB and PQ are considered to be equivalent (equal), if  projection of   AR on PA is equal to AB and projection of  PF on PP is equal to PQ

PrPAAR=AB,   PrPPPF=PQ,    AR = PF 

Equivalence of vectors AB  and  PQ defined in such a way is multivariant. However, if the surface S in GEN  is a plane,  the equivalence of vectors  AB and PQ is single-variant,

PrPPPF =PrPAAR  and hence AB = PQ in the space GEN. 

Equivalence of two vectors in different points must be defined, if  GR is the space-time geometry. Such equality is necessary for identification of like physical bodies in different places of the space-time.

Multivariance of the equivalence relation means its intransitivity. But intransitive equivalence relation is impossible in any logical construction. It means, strictly speaking, that one cannot speak about inconsistency of the Riemannian geometry, However, the supposition on single-variance of  the two vectors  equality does not take place in the Riemannian geometry.

It seems that you are concerned with holonomy, and gauge. This problem has been addressed by classical differential geometrists. 

Yuri ~

I’m not sure I understand quite what you are saying here. A Riemannian manifold doesn’t have to be thought of as embedded in a higher-dimensional Euclidean space. It has intrinsic geometrical properties. At every point there is a “tangent space” (a vector space, not a "plane") with basis dxi and the metric at that point is a scalar product  ds2 = gijdxidxj. Vectors at two different points A and B can be compared by “parallelly” transporting one of them along a curve C connecting A and B:  v i(A) → v ′ i(B) = v i(A) + ∫C {ijk} v jdx k. Then v ′ i(B) is dependent on the choice of the curve C. That’s what you are referring to as the “multivariance”, I think. But that’s not an inconsistency. On the contrary, it’s the basis of the definition of Riemann’s curvature tensor.