Alternative to Ricci flows
08 Apr 2009 18:19 GMT
Describing (smooth) geometry so that geometric descriptions are based on space-forms is a better way of describing geometry than describing geometry based on curved coordinates and diffeomorphism fiber groups.
Because of the stability of the hyperbolic space-forms the space-form based way of describing geometry makes it relevant to both classical material geometry (composed of stable atoms), and to the spectral properties of material which can be both macroscopic and microscopic, so that the microscopic descriptions include the spectral properties of quantum systems, and the macroscopic properties describe stable solar system structure.
The academic authorities are highly qualified at using a limited language and then limiting it further so the language can only describe patterns which are not useful. W Thurston’s conjecture about geometry being related to space-forms opens up the discussion about geometry, and the ideas expressed here broaden that context in a useful way.
One can define words in mathematics as one wants. If a coordinate metric space is metric invariant from point to point in the coordinate space in relation to the system’s fiber group, which can be either a general diffeomorphism group or, in this case, an isometry group, then one could say these spaces are homogeneous (invariant to translations, or displacements) and isotropic (invariant in all directions, which seems to imply rotational symmetry although no symmetry may be present) and this might be considered flat.
However, flat is often defined in regard to curvature (a second derivative of a metric function defined on a manifold, or the possible types of coordinate transformations of a metric-space, eg diffeomorphic or isometric) and in this case, only those metric-spaces whose curvatures are zero would be called flat.
The hyperbolic metric space is a metric space which is metric invariant and homogeneous and isotropic, yet it has a constant (scalar) curvature of (-1), so usually this is not called flat, so perhaps it would be better to call this space metric invariant, in regard to the isometry fiber group, SO(3,1) or SL(2,C), which are appropriately adjusted for acting on hyperbolic space. Note: Space-time, R(3,1), is (isomorphically) equivalent to hyperbolic space, H(3).
When considering interacting space-forms, where the space-forms model material then the interaction defines an interaction space-form which has both spatial and material (space-form) faces, and it is defined in the next dimension metric space (in regard to the dimension of the metric space which contains the material space-forms which are interacting). That is, the interacting material space-forms are 2-dimensional and their containing metric space is 3-dimensional and the greater containing metric space, ie the metric space which contains the (3-dimensional) interaction space-form, is 4-dimensional. The interaction space-form is changed, or deformed, with each spin rotation of metric-space states, ie states of the 3-dimensional metric space. This deformation is done by an action of the isometry group of the 3-dimensional [4-dimensional ?] metric-space on the spatial faces of the 3-dimensional interaction space-form. This deformation is usually a displacement in the radial direction between the two interacting material space-forms (though several material space-forms could interact), and this (radial) direction of change is due to the 3-sphere geometric structure of the SO(3) part of the SO(3) x R^3, isometry group for Euclidean space, so that the best identified direction of displacement (due to the interaction) is along the well defined radial direction on the 3-sphere transformation group in relation to 3-space, as opposed to the many tangent directions, which the sectional curvature of the interaction space-form (between the two interacting materials) defines (in some tangent direction, as opposed to the radial direction).
Sometimes the deformation of the interaction space-form (in the 4-dimensional containing metric space) differs from what one would expect from dynamical changes. This alternative type of deformation would be due the existence of critical ranges of both geometric size of the interaction space-form and the interaction space-form’s associated energy, where both size and energy considerations could be relevant for various ranges of values for each property (size, energy). The critical sizes are determined by the stable spectral properties of the containing metric space, which can be modeled as space-form. When these critical ranges of values are realized by the (interaction space-form) system then more rapid, and different deformations of the interaction space-form (or interaction manifold) can take place, so that these deformations are based on spectral resonances between the interaction space-form and the 4-dimensional metric-space’s spectra (so that the energy of the system is consistent with these metric-space spectral energy properties), where the 4-dimensional metric space is also modeled as a 4-dimensional space-form. There also exist properties of energy compatibility with a stable space-form system, into which the interaction space-forms deforms, and the spectral properties of the containing metric-space.
In other words the differential structure (or the fiber group) of the interaction space-form, or manifold, is related to the Lie algebra of the isometry group so that the isometry group has a discrete isometry subgroup structure which determine spectral resonances and affect the smooth properties of the manifold (or interaction space-form) deformations.
These manifold deformations could be modeled as metric function deformations, but they are contained in a metric space whose metric remains unchanged so that resonances and energy properties determine changes in spectral scale and spectral geometric organization so that the scale changes can be both larger and smaller spectra into which the deformations can take place. Furthermore, separate Euclidean and hyperbolic spaces are inter-related within the system (mostly) by means of resonances between the space-form structures which compose the inertial and charged material properties of the system.
It seems that this deformation could be equivalently be described as a metric-function deformation, so that the deformation depends on the spectral resonances that the containing 4-dimensional (space-form) metric space determine. Such a rapid deformation of an interaction space-form into a stable 3-dimensional space-form (in a 4-dimensional containing metric-space, so that the relevant stable parts can project down to the 3-dimensional metric-space) could be called a Ricci flow of a metric function deformation since the spectral structures, or sectional curvatures, of all the space-forms which compose the interaction space-form can change with the deformation. This is clearly a possibility in the above described context of interacting material space-forms in a multi-dimensional structure to space in which each dimensional level of space, itself, has the structure of a space-form. A theorem by H Coxeter about the properties of discrete (hyperbolic) groups, influences the discrete subgroup properties of the (hyperbolic) metric-spaces whose spatial dimensions are different from three. However, in the more fixed idea of Ricci flows hyperbolic space-forms can only get bigger, but in the new context this is not a necessity.
A Ricci flow is typically defined in the professional literature in a manner which does not allow such descriptions of material interactions, given above, to be relevant to metric deformations. If g is the metric function then the Ricci flow is defined as the heat equation, or second order parabolic differential operator, defined on a function space of metric functions.
This would be:
dg/dt=Lg where L is the generalized second order Laplace operator. For general metrics this would be dg/dt=L(0)g + Ricci + Rg, where L(0) is the generalized Laplace operator and R is the scalar curvature (sum of the sectional curvatures, or trace of the [diagonal] Ricci curvature). But L(0)g may not be zero as it is usually assumed in discussions about Ricci flows. L(0)g may be equal to some density function (eg mass, current, energy etc) or it could be related to a resonance set, or energy function, or spectral function identified by the discrete group of the isometry group (or a discrete subgroup of the differential structure of the manifold). In particular, L(0)g would not be zero if the differential structure (or the space’s fiber group) is an isometry group (or unitary Lie group) which has a discrete subgroup structure. So that the deformation of the metric function, dg/dt, could be partly dependent on the spectral (or discrete subgroup) structure of the metric space’s differential structure (or fiber group), or it could be related to interaction geometry and/or interaction energy (which might be related to sectional curvatures of the interaction space-form).
To relegate the deformation of a metric function which only depends on the extrinsic curvature of the containing unbounded coordinate space, is looking at the geometry of interaction (or of inertia) from a very narrow viewpoint.
This demonstrates how a Platonic truth within mathematics can be very irrelevant to discussion about existence, and very much only about formally adhering to rules so as to adhere to a competition and to a competition for prize moneys.
That is, the action of a heat differential operator on a metric function so as to show how a metric function flows within a metric function’s function space is not a good model of how a manifold (or its metric function) deforms during interactions. Neither the interaction, ie simply a metric function and its geodesic implications, nor the containing metric space (or domain space for the metric function) are properly modeled for what (might) actually exist.
Rather in an alternative model of interaction (or inertial changes) an interaction manifold deforms in a manner which is based on its natural dynamic interactions (or deformations), including its total interaction energy, and the spectral relation (or resonances) that the interaction manifold has to the manifold which defines its containing metric-space.
As usual the academic world is concerned about irrelevant Platonic forms (or Platonic truths) so that its models of existence are very poor, and its descriptions not useful or in error since they do not model the correct context of existence.
Apparently, the stationary points of the Ricci flows within a restricted idea about how a metric deforms in a metric-function function-space, ie the critical points of the heat equation (or heat deformation), ie dg/dt=0, are the metric invariant spaces of Euclidean space, spherical space, and hyperbolic space. The space-forms of these spaces are metric invariant and their geometries are based on separable (or locally perpendicular) coordinates.
Interactions are not determined by geodesics of well defined manifolds, rather they are sets of space-forms which form geometries in the context of their isometry group transformations so that the isometry groups have a discrete subgroup structure which determine the space-forms which they (the metric-spaces of the isometry groups) contain.
There are a number of issues which the Riemannian formulation of metric function deformation does not deal.
(1) a Riemannian manifold is essentially a locally Euclidean space, thus it does not naturally contain hyperbolic spaces, and the proper dimension for a Riemannian manifold, which contains a general manifold of dimension n, is 2n.
(2) Both the metric function and the Ricci tensor can be represented as symmetric matrices and thus can be diagonalized so that the Ricci tensor has eigenvalue structures. Thus the heat equation can be written term by term as dg(i)/dt=k Ricci(i) or as an eigenvalue equation, where it could be that Ricci = h(x) g (i), so that h(x) is a scalar function of position.
In three dimensions this could be interpreted in terms of the metric function flowing to the sectional curvatures of the planes perpendicular to the (i)-directions on a (the) manifold (or space-form).
This could be (is) similar to d d g = energy spectra of the metric space, or d d g = k(i) g, ie the spectral (energy) structure of a metric-space, associated to metric invariance. This could lead to the idea of space-form geometries being related to manifold structures, and this could be the ideas which motivated W Thurston to introduce his geometric conjecture that “all (3-dimensional) manifolds are composed of pieces of metric invariant geometries or space-form geometries.” Thurston’s work broadens mathematical discussion, usually math discussion becomes more narrow and specialized.
Nonetheless Thurston’s discussion about geometry can be broadened, perhaps these geometries need to be placed in a greater containing context of multiple dimensional metric-space levels, so that their interactions are determined by displacements in space (displacement deformations of the interaction space-forms) and their geometric deformations (of the interaction in relation to its forming a stable system) are determined by resonances and energy levels in relation to isometry groups with particular discrete group structures.
Truth is about the correct context and the resulting usefulness of a verifiable description. Why limit truth to useless Platonic truth, ie truth within a fixed language? A fixed language which has great limitations as to what it can be used to describe, thus limitations on its usefulness. The context of a descriptive language needs to always be changed if one wants a (changed) descriptive language to be ever more useful.
How is geometry to be determined so that it is in its proper context, so that it is useful in the description of existence.
Should the descriptions of geometry deal with:
(1) Manifolds and diffeomorphism groups,
(2) Metric functions and their isometry groups, or does
(3) classical gravitation and electromagnetism models of material geometry and inertia in space, or are
(4) the descriptions of quantum systems (or equivalently spectral systems), which are described as sets of Hermitian differential and position operators acting on function spaces, so that the function space determines probabilities of spectral states (or events), and hence these ideas oppose material geometries (by the uncertainty principle), or
(5) should spectral and interaction properties of material in space be described with discrete isometry subgroup or space-form structures?
Thurston’s ideas suggest that manifolds, that have diffeomorphism groups, can all be pulled apart into separate space-form structures so the diffeomorphisms transition from one space-form (or metric invariant) type to a flat type to yet another space-form type.
Is it a…:
manifold and diffeomorphism groups, or an
invariant (flat) metrics and their isometry group and its metric-space states’ covering unitary group, or
do both stability and interaction properties to be the things (or patterns),
…..which need to be the basis for geometric description of existence?
If manifolds with general metrics are the basis for geometry then the idea of molding a spherical surface so as to transition between space-form pieces of the manifold takes the dominant position in the descriptive language.
Is the main property (or function) of geometry?:
(1) the shape of space (or metric deformations [or changes] from point to point on a manifold which has a metric function defined on itself) and subsequently is a description of interaction about inertia on a curved space, or
(2) is the main property about stability so that classical geometries can exist in metric invariant spaces?
Are the geometric properties of existence primarily about both:
(1) the stability of material and space, that is, the spectral and orbital properties of geometric-spectral systems of all different sizes from microscopic to macroscopic, or
(2) the interaction structures and properties of resolving properly energetic interacting systems into spectral structures which are mostly determined by resonances of the new system with the metric space within which the new system is contained?
Group transformations of a system’s metric function are either diffeomorphic or metric invariant.
There are three main types of invariant metric space structures and these spaces have constant curvatures of: zero, one, and (negative one) respectively:
Note: Spherical geometry and its metric function fit nicely into Euclidean geometry where the spherical metric is the Euclidean metric constrained to the sphere.
W Thurston has conjectured that all (3-dimensional) manifolds can be broken apart (or cut) into pieces (where the cuts occur along toral shapes) which are space-forms of the discrete groups of: Euclidean, spherical, and hyperbolic spaces.
But within what space are these manifolds contained? If inertia can only have properties of spherical symmetry in 3-Euclidean space then one can conjecture that the geometric manifolds of Thurston are contained in space-form structures of Euclidean and hyperbolic spaces (and/or space-form structures of metric invariant spaces) of higher dimensions and of different types of metric function signatures. This would necessitate that a containing metric space have spectral properties so that these spectral properties determine the type of stable spectral material structures which such metric spaces can contain.
The different signatures for the different dimensional metric spaces (or their space-form models) result in the metric-space being divided into spatial subspaces and time subspaces, where the higher dimensional time subspaces are related to the energies of new “physical” systems composed of new materials in the higher dimensions. That is, in 3-dimensional space there are inertial properties related to Euclidean space and charged material properties related to space-time (or hyperbolic space). However, in R(4,2), a new material will be defined which forms systems and those systems will have energy and that energy is related to one of the two time dimensions, the other time dimension will be related to the energies of the charged geometries within R(4,2).
One should note that this is a much more diverse setting for geometry and its relation to the discrete subgroup structure of their isometry fiber groups.
The case in which the Poincare conjecture was supposedly proved is a space with much less possibilities for geometric structure, and interaction structure than the limited molding of clay on surface structure of the Ricci curvature tensor, because the Ricci curvature structure has not been applicable in any meaningful way to the description of physical geometry and its subsequent interactions, other than the essentially irrelevant context of big bangs and black holes if either of these properties actually exist.
The Poincare conjecture and the prizes which surround such statements show the limited viewpoint which the experts create for themselves, and it is a good example of why intellectual oligarchs are bad for considering truth, in relation to how such prize problems determine personal worth of mathematicians (which is derived from winning these contests).
The Poincare conjecture is a pattern which is not easily accessible within the fixed descriptive language of math orthodoxy. Thus the experts consider it to be a good point about which to base a contest, a prized problem, but this focus of attention has become a narrowing force in mathematics and its value is irrelevant within the bigger context of existence which might be best described in the context of discrete isometry subgroups in a higher multi-dimensional-spectral hierarchy which seems to be a more relevant basis for a language to be used for describing existence.
Material does seem to determine geometry, but the geometry is stable and such stability is better modeled as the spectral properties of hyperbolic space’s space-forms. Furthermore, the metric-space which contains the material geometry is (also) better modeled as a space-form. Within this context material space-form interactions take place and can often result in new stable space-form structures, so that this process depends on the spectral properties of the metric space which contains the interacting system.
If one assumes that material and space can both be modeled as space-forms in a multi-dimensional spectrally dependent context then existence can be described in a useful manner, more useful than what is now being described. This context supports the idea of Thurston. Perhaps Thurston’s ideas are better used as axioms rather than trying to prove them in a language within which they might not be provable, but the space-form basis for describing geometry is more useful.
The academic authorities are highly qualified at using a limited language and then limiting it further so the language can only describe patterns which are not useful.