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# FROM R4 CURVILINEAR SPACE TO THE CYCLE SPACE

Olivier Maurice

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Olivier Maurice. FROM R4 CURVILINEAR SPACE TO THE CYCLE SPACE. 2024. hal-04448080

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# FROM $\mathbb{R}^4$ CURVILINEAR SPACE TO THE CYCLE SPACE

Olivier MAURICE, AFSCET

17 Janvier 2024

## 1. INTRODUCTION

In [1] we have submitted to consider Kron's connectivity as a metric wearing the links between the $\mathbb{R}^4$ space and the cellular topology including cycles, edges and nodes spaces. In this article we deeply study this proposal and its consequences. This work comes as a complement for introducing [1].

## 2. DEFINITION OF A CYCLE BOUNDARY

Let's consider a face $\Phi \in \mathbb{R}^3$, defined by the set of points $x_i/\Phi = \{x_i\} / \forall (i, j) \in \mathbb{N}^2$, $|x_i - x_j| < \epsilon, \epsilon > 0$. We note $\phi = \mathbb{R}^3 \setminus \Phi$. Then a cycle $C$ is obtained making $C = \{x'_i\} / \{x'_i\} = \phi \cap \Phi$.

## 3. CYCLE SPACE

The common space is traveled by curvilinear trajectories $u_k = \{x_i \in \mathbb{R}^3 / \forall i | x_i - x_{i+1}| < \epsilon, \epsilon > 0\}$. The neighborhood constraint on $u_k$ is reduced to a constraint based on two neighborhood points. In the case of the neighborhood in $\Phi$, it concerns any couple of points in $\Phi$, not only consecutive points. An edge $S_k$ can be seen as a continuous part of $u_k$:

$$ (1) \qquad S_k \subset u_k, \quad S_k = [x_\alpha, \dots, x_\beta] / \exists (x_i, x_j) \in u_k, x_i < x_\alpha, x_j > x_\beta, S_k \cap u_k = [x_\alpha, \dots, x_\beta] $$

We can attach a flux vector to $S_k$: $J^k$ which boundary is $(x_\alpha, x_\beta)$ describing any particle movement along $S_k$, i.e. going through the points $[x_\alpha, \dots, x_\beta]$. For a trajectory $u_k$, it's impossible to have a contact point $x_j$ with $|x_i - x_j| < \epsilon$ and $j \neq i + 1$ or $j \neq i - 1$. But a trajectory can be as twisted as it wants. By the fact, for creating a face boundary, we need at least two trajectories $u_k, u_m$ crossing each other on two nodes (two points). This can be written $\{x_1, x_2\} \in (u_k, u_m)$. In other words: $u_k \cap u_m = \{x_1, x_2\}$. The couple $(x_1, x_2)$ defines on $u_k$ and $u_m$ two edges $S_k$ and $S_m$ so that:

$$ (2) \qquad \left\{ \begin{array}{l} S_k \in u_k, \quad S_k = [x_1, x_i, x_2] \\ S_m \in u_m, \quad S_m = [x_1, x_j, x_2] \end{array} \right. $$

with $\forall i, j, x_i \neq x_j$. These two edges wear two flows $J^k, J^m$ and constitute a face $\Phi_\alpha$ boundary with $\Phi_\alpha \cap (\mathbb{R}^3 \setminus \Phi_\alpha) = \{x_1, x_i, x_2, x_j\}$. Let $\Phi_\alpha \cap (\mathbb{R}^3 \setminus \Phi_\alpha) = \partial\Phi_\alpha$, $\Phi_\alpha$ boundary identified with a scalar $a_\alpha$. We can measure the flows $J^k, J^m$ on the $\Phi$ boundaries base and write: $J^\sigma = g^{\sigma\alpha}a_\alpha$. $g$ appears as a metric.
## 4. FLUX

The flux is identified by:

* the edge that wears the flux;

* a complex amplitude, speed function.

The flux measurement is automatically given by the metric $g$. The boundary $\partial\Phi_\alpha$ wears also a closed flux $a_\alpha$. We have: $J^k \leftarrow (S_k, s^k)$, $S_k$ the edge being the geometrical description of the flux and $s^k$ being its complex amplitude. Similarly we have: $\check{a} \leftarrow (\partial\Phi_\alpha, a_\alpha)$. A $s^k$ measurement can now be written:

$$ (3) \qquad s^k = g^{km} a_m $$

$s^k$ browsing the edge $S_k \in \partial\Phi_\alpha$. By this way all seems to be well defined. We retrieve our introduction in [1]. We can develop forms like $a_1 = S_k u_k + S_m u_m$, all the equations on $a_i$ making the function vector $\varphi$. The base $b_k = \partial\varphi/\partial u_k$ leads to the metric $g_{km} = \langle b_k, b_m \rangle$. Its inverse $g^{mk}$ gives the measurements of the edge depending on the cycles.

## 5. BOUNDARIES

We imagine a set of faces $\{\Phi_1, \Phi_2, \dots, \Phi_n\}$. These faces follow one another, i.e. $\exists(i, q)$, $\partial\Phi_i \cap \partial\Phi_{i+q} \neq 0$, $i \in [1, \dots, n-1]$. We have to define the flows orientation. An edge $S_k$ being an ordonned set of points $\{x_i, \dots, x_j\}$, it wears a flux $J^k$ going from one edge extremity to the other (the edge boundary having two nodes attached to the edge extremities), so here going from $x_i$ to $x_j$. The flux complex amplitude $s^k$ can be defined by:

$$ (4) \qquad s^k = f(\overrightarrow{x_i x_{i+1}}, \overrightarrow{x_{i+1} x_{i+2}}, \dots, \overrightarrow{x_{j-1} x_j}) $$

Also $a_\alpha(\partial\Phi_\alpha)$ is oriented. If $\partial\Phi_\alpha = \{x_1, \dots, x_n\}$ with $x_{n+1} = x_1$, then:

$$ (5) \qquad a_\alpha = f(\overrightarrow{x_1 x_2}, \overrightarrow{x_2 x_3}, \dots, \overrightarrow{x_n x_1}) $$

Under these definitions: $\partial\Phi_i \cap \partial\Phi_{i+q} = S_k \Rightarrow s^k = g^{k\sigma} a_\sigma = \pm a_i \pm a_{i+q}$. $S_k$ is the common boundary to $\Phi_i$ and $\Phi_{i+q}$. $g^{k\sigma}$ is equal to 1 if the orientations of $s^k$ and $a_\alpha$ are the same. $g^{k\sigma}$ is equal to $(-1)$ if $s^k$ and $a_\alpha$ are opposites. Both curves (the edge and the cycle) are completely described using a curvilinear location $u_k$ which takes consecutively the origin of each elementary vector as location in $\mathbb{R}^3$ (for example, $u_k = x_1$) then moves to the next point following the elementary vector direction (for example $\overrightarrow{x_1 x_2}$).

## 6. EXTENSION TO DEFINE REAL OBJECT SUPPORT

All the previous concepts can be extended considering the forms $J^k$, $\check{a}_\alpha$ as manifolds. Using homotopies [2], we see the $\check{a}_\alpha$ as closed surfaces being boundaries of volumes $\Phi_\alpha$. these surfaces have a thickness $\epsilon$ and an orientation linked with the first flux eigen-mode on $\partial\Phi_\alpha$. The flows $J^k$ become slicks having the same thickness than the $\check{a}_\alpha$. They belong to curvilinear ribbons $u_k$. The thickness are many times less than the lengths and widths of the $u_k$. The guiding lines of ribbons, slicks and closed surfaces are the trajectories, edges and cycles previously studied.
These equivalences justify of the real object representations through cellular topologies and
graphs.

The graph boundary (or skin) can be defined as the set of edges {$J^k$} for which only one
component of $g^{(k)\sigma}$ (i.e. the $k^{eme}$ line of $g^{k\sigma}$) is equal to $\pm 1$.

7. THE TIME DIMENSION OF {$\check{a}_{\alpha}$} AND STRINGS

$\check{a}_{\alpha}$ having a complex amplitude, its phase gives the classical time flow. The $u_k$ network
supports also the field space. Electromagnetic couplings can occur between edges or cycles
passing through the field space. Measurements in the wide meaning of the word translate
the interactions between the physical space described by the cell complex network and the
electromagnetic field space. Some trajectories $u_k$ follow the field distribution and can go from
one cycle to another, or from one edge to another. These kind of interaction can be generalized
to functions linking a component of the source vector to a component of the cycle current.
These functions not supported by edges or cycles are called strings[3]. Strings do not change
the network Poincaré's characteristics. An edge or a cycle current interacts with an underlying
layer supporting the electromagnetic fields. This interactions can be seen as a measurement.
In another space location, another measurement process couple the field energy with a local
edge or cycle. In these interactions, a delay accompanies the coupling, translating the process
causality. Each edge belonging to a cycle, the cycles phases date the temporal events associated
with the measurements. The concept of time is above all a perceptible concretization of these
events.

The layer where the interaction propagation occurs is an abstract one, not constrained to
follow locally straight trajectories like the particles. It's a space of waves that constitutes
the quantum space or the relation space. It is never accessible directly, but only through
measurements. In this space, the time doesn't exist. It appears through the events associated
with what we called in general measurements.

For physical interactions, we may call the abstract space the bosonian space {$\varphi_k$} while the
curvilinear space $u_k$ may be called the fermionian space.

The grid $u_k$ is drawn on a continuous sea $\equiv (\mathbb{R}^3, \mathcal{C})$ which is the abstract space.

How may we define more accurately the relation between both spaces $\mathcal{C} = \{u_k\}$ and $\mathcal{O} = \{\varphi_k\}$?

Obviously, the two spaces are topological spaces. $\mathcal{O}$ is an infinite continuous space. We can
define a map $Q$ that can be seen as an homotopy lifting: $Q : \mathcal{O} \to \mathcal{C}$. Using a differentiable
manifold $K$, we define two maps $f : K \to \mathcal{C}$, $\tilde{f} : K \to \mathcal{O}$. $Q$ is called a fibration[4] if any
homotopy $F : K \times I \to \mathcal{C}$ accepts a restoring $\tilde{F} : K \times I \to \mathcal{O}$ ($K$ being of dimension $n$,
$I$ is in our case a $n-1$ dimension manifold). It means that any construction in $\mathcal{C}$ resulting
from subspace product in order to make for example a closed surfaces starting from a cycle
can be continuously translated in extension of the set of points in $\mathcal{O}$. This is an intrinsic
property of Maxwell's relations in particular the one giving the potential vector field definition
depending on the current density. Developing the current through a current density spreading
is immediately translated in a continuous field distribution pressed on the new slick definition, coming from a homotopy applied to the previous definition.

The space $\mathcal{O}$ is called *base*, and $\mathcal{C}$, *fiber space*. $\mathcal{Q}$ is a *projection*, the whole process of relation between $\mathcal{O}$ and $\mathcal{C}$ being a *homotopic connection in the fibration* $\mathcal{Q}: \mathcal{O} \to \mathcal{C}$. The image maps $\mathcal{Q}^{-1}$ are *fibers*. These fibers are the $u_k$ skeleton.

The concept of fibration is a main concept in physics and for our application. Let's read another description of these fiber spaces given by Penrose [5].

$\mathcal{O}$ is the base space, $\mathbb{R}^4$. In our opinion, $\mathcal{O}$ is the complex space associated with quantum electrodynamics, so $(\mathbb{R}^3, \mathbb{C})$. $\mathcal{F} = \{u_k\}$ is the fiber space: our physical and topological cellular complex. As said by Penrose[5], $\mathcal{F}$ is made of as many fibers as $\mathcal{O}$ accepts. The trajectories $u_k$ become the fibers and $\mathcal{F}$ is the set of $\mathcal{O} \times u_k$ defining the ribbons and as a consequence, the closed surfaces then the slicks. The set of $\mathcal{F}$ are the couple of points $(a, b)$ where $a \in \mathcal{O}$ and $b \in u_k$. What interests us is more precisely the couples $(A, B) \in \mathbb{C}^2$ defining a fiber space of Clifford-Hopf [5].

If we consider the vector potential $\mathbf{A}$ with $\mathbf{A} \in \mathcal{O}$. Along any current flow we can imagine a plan, perpendicular to the flow and being a vector potential wave plane $P$. If the ribbon $u_k$ wears the current, it appears as a fiber $U$ and $\mathcal{F} = P \times U$ describes Maxwell's equation. Now making intersections between $u_k$ leads to closed circulation $\check{a}_\alpha \in \mathcal{F}$ for which $\mathrm{div}\mathbf{A} = 0$. Coulomb's gauge is the fundamental gauge of $\mathcal{O}$. The electromagnetic flux $a_\alpha$ is defined by:

$$ (6) \qquad a_{\alpha} = \int_{\partial} \partial \Phi_{\alpha} \cdot \mathbf{A} $$

which gives the way for measuring the currents $J^k$: that's where we start.

## 8. THE SCALE CONDITION

To be able to make measurements of edge flows through the metric $g$ supposes that the edge flux can be projected to a single scalar. This condition supposes that the edge length remains short compare to the wavelength. It means that the minimum wavelength that can be associated with one topology is directly defined by this topology. When propagation phenomenon intervene at high frequencies, this propagation can be taken in charge by the abstract space, the limit conditions where the signal is created or received being cycles. These two cycles can be also seen as measurements.

The first condition comes from the assumption of homogeneous flux in $\check{a}$. Noting $A^0 = \psi$, the scalar part of $\mathbf{A}$, if there was propagation along $\partial\Phi_\alpha$, this implies that:

$$ (7) \qquad \nabla \times \overrightarrow{\mathrm{grad}\psi} \neq 0 $$

In that case, the development for transforming Kirchhoff's equation into Kron's one in no more valid and all the developments realized in [1] are out of context.
The scale condition at the cycle level is a fundamental assumption in the configuration space taken as input for the tensor analysis of networks, and allows all the topological relations developed in [1] and here.

## 9. GRAVITATION

In $O$, the time must be replaced by the wavelength $(x, y, z, \lambda)$. In relativity, the wavelength transforms as:

$$ (8) \qquad \lambda' = \lambda_0 \left[ \frac{1 - \beta \cos(\alpha)}{\sqrt{1 - \beta^2}} \right], \quad \beta = \frac{V}{c} $$

$\alpha$ is the angle between the line joining the observer and the mobile and the mobile movement speed. In the case of a circular motion, this relation becomes:

$$ (9) \qquad \lambda' = \gamma\lambda_0, \quad \gamma = \sqrt{1 - \beta^2}^{-1} $$

Under these conditions, Lorentz's transformation becomes:

$$ (10) \qquad \left\{ \begin{array}{l} \theta' = \gamma (\theta_0 - \beta\lambda_0) \\ \rho' = \rho_0 \\ z' = z_0 \\ \lambda' = \gamma (\lambda_0 - \theta_0/\beta) \end{array} \right. $$

Applying this transformation to the static gravitational field creates a "gravirotational" field[6]. Both longitudinal and transverse gravitational fields curve the space. Considering the mass energy of a photon $h\nu/c^2$ ($\nu$ is the photon frequency), the trajectory followed by the photon is the light line and defined the space metric $u_k$. In this story, the time is a measure obtained by:

$$ (11) \qquad \frac{1}{\omega} \int_{x_1}^{x_2} dx \psi^* \left( \frac{1}{i} \frac{\partial}{\partial x} \right) \psi = \delta t $$

The space variation of the wave defines the time. We don't know how to measure the time. What we can measure in fact is just some time interval.

The gravirotational field for a rotating mass $M$ at angular speed $\Omega$ is[6]:

$$ (12) \qquad \mathbf{h} = -G \frac{M\Omega}{rc^2} \mathbf{z} $$

The corresponding potential is:

$$ (13) \qquad \mathbf{A} = -G \frac{M\Omega}{c^2} \theta $$
The classical gravitational force acts between the masses while the relativistic one acts between movements of masses. Once defined the interactions between little edges of movements, we replace the edges currents by the cycle ones: this gives the way for resolving the problem.

Following Kron's approach we can describe the problem through static particles in the edge space. Then we express these edges depending on cycles through a metric that depends on the angle. The time deviation in Newton's law creates Christoffel's coefficients that lead finally to Einstein's equations[1]. These equations define the light trajectories connected to the {$u_k$}, they are the directions followed by the electromagnetic field and define the wave planes $O$, base of the fiber space $F$.

We can imagine a mass $M$ around which a photon travels[7]. This mass is located in a cartesian space of base ($x^1, x^2, x^3$). The photon is obliged to travel on a ribbon attached with a cylindrical coordinates (for example) ($y^1, y^2, y^3$). Between each system of coordinates exists a vector function taking in charge the change of bases :

$$ (14) \qquad x^i = f(y^j) $$

A curvilinear base can be defined:

$$ (15) \qquad \mathbf{b}_k = \frac{\partial f}{\partial y^k} $$

Starting from this base, we can define the fundamental metric $G_{ij} = \langle \mathbf{b}_i, \mathbf{b}_j \rangle$. The kinetic energy of the photon is given by:

$$ (16) \qquad T = \frac{1}{2} \left( \frac{h\nu}{c^2} \right) G_{ij} v^i v^j $$

We know that the force comes from a potential, including the gravirotational one created by a local relativistic effect:

$$ (17) \qquad V = -G_0 \frac{M}{c^2} \left( \frac{h\nu}{r} + \mathbf{\Omega} \cdot \mathbf{\theta} \right) $$

The photon trajectory is obtained resolving Lagrange's equations:

$$ (18) \qquad \frac{d}{dt} \frac{\partial T}{\partial \dot{y}^i} - \frac{\partial T}{\partial y^i} + \frac{\partial V}{\partial y^i} = 0 $$

Solving these equations gives the photon's trajectories, and these trajectories define the wave plane movements creating the bases of the fibers $u_k$. It remains one difficulty in this development. As we evolve in $O$, we would expect that the time does not appear in these expressions. In fact the source coming from $V$ can be easily transform in their Laplace's expressions. Applying the rule $k(u.v) = ku.v + u.kv$, $k$ being Laplace's operator, and $kG_{ij} = \Gamma_{ij,\alpha}\dot{y}^\alpha$. After all computations made, once the speed spectrum $X(k)$ are known we can compute the quasi-peak value of the speed making:

$$ (19) \qquad \bar{v} = \sqrt{\int_V d\nu X f X^*} $$
Note that it is similar to a measure. *f* is a form factor approximated by *TD*/*τ*, *TD* is the temporal domain considered (linked with the first mode of the space studied, *D* = *λ*₀/*c*), while *τ* is the source duration associated with the lowest mode of the source.

10. FROM THE REAL OBJECT TO THE EQUIVALENT GRAPH AND POLYHEDRA (CELLULAR COMPLEXES)

Often, engineers find difficult to go from a real object to its representation by a graph. Once we have the wavelength/system dimensions ratio in mind, this helps to identify possible system parts that can be associated with edges. The simplest way perhaps for making this link consists in the identification of energy flows. Each subvolume where the energy remains constant constitutes a part. Depending of the kind of energy, Lagrange's equations lead to dissipative impedance (*R*), capacitive one for stocking (*C*) or inertia and inductance one (*L*). When constant structures behave like waveguide, their equivalent model uses the wave space. That's why the first study to make before any modeling work is to identify the dimension to wavelength ratio. Long structures become modeled using two cycles for their limit conditions. It is clear that the lower the wavelength, the higher the number of parts for the same system. A static part without any propagating structure has for limit frequency the first mode for which *L* = *λ*/2. Down to this limit, the part associated with a single edge must be represented by a couple of cycle. If we decrease again the wavelength, we reach a new breakup where we lost the part structure that appears as a chaotic mixture of molecules. At this level the graph is a chemical one and translates the molecules movements and links. But whatever the scale studied, the part models always come from a Lagrange's equation declination.

11. CONCLUSION

Our universe is a bosonic or relation space over which a curvilinear grid is placed. Any
system made of subparts can be described as a group of sets of points managed through a
complex cellular topology. In particular, set of points that can be assimilated with edges can
be measured depending on cycles fluxes that can be constructed starting from the grid. This
whole structure serves of base for the tensor analysis of networks described in [1].

Two spaces appear as two layers inseparable: one $\mathcal{O}$ is an abstract space, unreachable, where quantum fields and abstract concepts as information can be defined. The second $\mathcal{C}$, is the perceptible space construct on $\mathcal{O}$. Measurements can be seen partly as homotopy lifting applied on $\mathcal{O}$ and defining objects in $\mathcal{C}$, components of a cellular topology (cellular complexes).

The fibers $Q^{-1}$ give the facility to construct the curvilinear trajectories $u_k$ on which all the cellular manifolds are defined. Two metrics are involved, a first $G$ gives the facility for measuring the locations on $\mathcal{O}$, and a second $g$ gives the facility for measuring the energy flux characteristics in a fermionian space represented by $\mathcal{C}$.

The measurement interactions are the key bridges for going from one space to the other.

We hope that with these added notions and first topological tracks, [1] should be better
described on its fundamental basis.
REFERENCES

[1] Olivier Maurice. INCLUSION OF THE TENSOR ANALYSIS OF NETWORKS INTO DIFFERENTIAL GEOMETRY. 2023. (hal-04286165)

[2] Allen Hatcher. ALGEBRAIC TOPOLOGY. Cambridge University press.

[3] Maurice, O., Reineix, A., Durand, P., & Dubois, F. (2014). Kron's method and cell complexes for magnetomotive and electromotive forces. arXiv preprint arXiv:1412.0187.

[4] Doubrovine, Novikov, Fomenko, *Géométrie contemporaine*. MIR edition.

[5] Penrose, R. (2004). The road to reality, a complete guide to the laws of the universe.

[6] J.-M. Rocard, *Newton et la relativité*. Que sais-je collection. OYF editor, 1986.

[7] D.Papin, A.Kaufmann, *Cours de calcul tensoriel*. Albin Michel edition, 1966.