Non-trivial topology in the analysis of energy and exergy of photovoltaic systems: a Riemannian approach

Abstract

The analysis of energy generation variation throughout the year reveals a geometric structure intrinsically linked to the solar analemma, whose patterns resemble the Möbius strip and Klein bottle. This behavior suggests that the global effective incident energy and the energy converted by solar panels do not follow a trivial linear relationship, but a distribution influenced by unconventional topological properties. Modeling this system in a Riemannian space—using the analemma’s metric and scalar curvature—allowed the formulation of a differential equation governing the generated energy, incorporating seasonal variations and dependencies on panel inclination and orientation. From an exergy perspective, a correspondence between exergy destruction, owing to system irreversibilities, and geometric effects, described by scalar curvature, was identified. The resulting equation, expressed through the Laplace–Beltrami operator, indicated that energy variation can be described in a space with non-orientable properties, supporting the hypothesis that the energy distribution behaves analogously to complex topological surfaces. This approach enhances the understanding of the relationship between solar energy generation and spatiotemporal geometry while providing a mathematical framework to optimize energy harvesting throughout the year. By linking the curvature of the analemma with exergy principles, this study proposes a new paradigm for modeling energy efficiency based on geometric and topological analysis.

Keywords
energetic topology; Riemannian exergy; analemma geometry

Introduction

The increasing demand for renewable energy sources has driven significant technological advancements, particularly in solar photovoltaic (PV) power generation. The need to reduce carbon dioxide (CO₂) emissions and minimize the environmental impacts of fossil fuels has positioned solar energy as a viable and sustainable alternative (Basem et al., 2025; Junior et al., 2025). The installation of large-scale PV systems has expanded globally, accompanied by a reduction in their costs, encouraging their adoption in several regions (Abdelaal & El-Fergany, 2023).

However, the energy conversion efficiency of solar panels depends on multiple factors, including module inclination and orientation, season, and installed load capacity, making it essential to analyze these variables to optimize the performance of solar plants (Rosso et al., 2022; Zhu et al., 2025; Li et al., 2024a; Khan et al., 2025). One of the main aspects is the inclination angle, which must be adjusted according to the latitude to maximize solar radiation capture throughout the year (Abdelaal & El-Fergany, 2023). Most installations use fixed configurations; however, periodic adjustments—monthly, seasonal, or annual—are recommended to match the variation in the solar trajectory and maximize electrical generation (Bao & Bao, 2024; Zhu et al., 2025).

In addition to inclination, studies by Mahmud & Kurtz (2024) and Arbaoui et al. (2025) have shown that the module orientation also directly impacts efficiency, as panels positioned perpendicular to the incident radiation significantly increase energy capture (Li et al., 2024). The influence of season on power generation is also considerable, given the variations in solar altitude and day length (Khan et al., 2025). Another crucial factor is the sizing of the inverter and the compatibility of the installed capacity with local conditions, as underscored by Rosso et al. (2022).

To understand and quantify the efficiency of PV systems, energy and, in particular, exergy analyses have been widely used (Rao et al., 2025; Nascimento et al., 2020). The exergy approach enables a more accurate assessment of the efficiency of PV systems Almas & Sundaram (2025). While energy analysis is based on the first law of thermodynamics, exergy is based on the second law, allowing the evaluation of useful energy that can be converted into work and the identification of losses due to irreversibility in systems (Chow, 2012). Energy and exergy analyses are widely employed to understand the efficiency of PV systems, as they provide a detailed view of energy conversion in solar panels (Almas & Sundaram, 2025). This approach is essential for a more accurate diagnosis of the performance and optimization opportunities of solar plants (Nascimento et al., 2020).

The apparent position of the Sun at fixed times throughout the year emerged as an intrinsic geometric manifestation of the distribution of energy in space–time, prompting the consideration of a new approach with a non-trivial topology (Dendrinos, 2022). Elliott et al. (2021) showed that the Sun does not return to the same position in the sky every 24 hours. For example, at true solar noon, its elevation gradually increases as one transitions from the winter solstice to the summer solstice. Solar days, defined as the time interval between two consecutive solar half days, are not of constant length, which causes an apparent shift of the Sun, now east, now west, relative to its position at the same time the previous day. When this variation is recorded over the course of a full year, it describes an "8"-shaped curve in the sky, known as an analemma.

The current work proposes an innovative approach by treating solar energy variation not only from a technical perspective but also as a phenomenon governed by complex geometric and topological structures. The analysis of the incident solar radiation on surfaces inclined at 10° and 20°, with different orientations (north, south, east, and west) throughout the four seasons, revealed that the scattering graph of the generated energy adopts the characteristic shape of the solar analemma, a figure traditionally associated with the apparent position of the Sun in the sky at fixed times of the year. However, here, it is reinterpreted as a geometric manifestation of the energy distribution in space–time (Dendrinos, 2022).

Studies such as those by Raza & Avila (2024), who accurately describe the solar trajectory using classical methods of Cartesian geometry, and Zhang et al. (2021a), who proposed continuous three-dimensional vector representations to visualize solar motion over time exist; however, this study advances their works by suggesting that generated solar energy can be treated as a tensor field on a deformed Riemannian manifold, integrating concepts such as scalar curvature, differential operators (such as the Laplace–Beltrami), and non-orientable topological properties.

This view is supported by Rapoport (2013), who proposed the "logophysics of the Klein Bottle," as a physical–philosophical model for dynamic, self-referential, and nonlinear systems, applicable to cosmology, geophysics, biology, and sensory perception. The Klein bottle, like the Möbius strip, is not only a mathematical construction, but also a paradigm that allows us to understand energetic structures with torsion and non-orientable continuity.

According to Yu et al. (2025), the growing energy demand and environmental impacts caused by the intensive use of fossil fuels have driven the search for renewable sources, with solar energy standing out for its abundance and sustainability. Although solar thermal systems have high efficiency and low emissions, their performance is still compromised by seasonal and climatic instabilities. To address these limitations, this study proposes an innovative approach based on differential geometry and topology, interpreting the variation in the generated energy as a spatially curved and non-orientable phenomenon, such as the Möbius strip and Klein bottle surfaces, with the goal of improving the modeling and predictability of solar systems.

Thus, as pointed out by Li et al. (2024a), the energy variation is interpreted as resulting from an energetic space endowed with torsions and topological properties analogous to those present in a Möbius strip. According to Li et al. (2024b), in the chemical theory of graphs, topological indices, such as energy, perfect marriages, and generator trees, have been widely applied in the description of hexagonal networks and in the modeling of complex structures. In this context, the Möbius strip—a non-orientable surface with only one side and a continuous edge—has been used as a structural metaphor for understanding complex biological systems, such as cancer progression and treatment, due to its ability to represent nonlinear, self-referential, and multifaceted dynamics. Similarly, in this study, using the same topological structure is proposed to interpret the variation in solar energy generated throughout the year, whose graphic patterns, especially the analemma, reveal geometric properties that transcend the classical Euclidean description. The distribution of solar radiation and its energy conversion, when analyzed on inclined surfaces and under different seasonal orientations, manifests an organization that refers to the characteristics of the Möbius strip: torsion, continuity without global orientation, and cyclic recurrence.

Similar to the study of oncology, topology is used to understand the intrinsic complexity of an unstable and adaptive biological system (Yanchu & Rong, 2024). His work proposes that non-orientable geometry can also offer an effective model to capture the curved, dynamic, and non-trivial nature of solar energy on the Earth's surface. This approach, based on surfaces such as the Möbius strip and Klein bottle, makes it possible to integrate seemingly distinct systems, such as cancer and solar radiation, under a common topological logic, highlighting the unifying potential of mathematics in modeling complex natural phenomena.

In its formulation, the Klein bottle is not merely a mathematical construct, but a fundamental physical model for describing dynamic, self-referential, and morphogenetic processes in cosmology, geophysics, biology, and sensory perception (Koam et al., 2023 and Gonçalves et al., 2025).

Thus, by associating the behavior of the generated energy with the curvature of the energetic space, this study proposes a new interpretation of solar radiation: not as a mere scalar and linear flow, but as a dynamic field that manifests complex geometric and topological patterns. This approach expands the possibilities for optimization in solar energy harvesting and provides a new perspective for understanding the natural patterns of energy distribution in complex systems.

Material and Methods

Analema solar

According to Pochont & Sekhar (2024), the analemma constitutes a celestial curve of octave configuration that denotes the apparent trajectory of the Sun in the firmament throughout the year, when measured at the same time and from the same observation point daily. This phenomenon emerges from the interaction between the Earth's axial tilt, which forms an angle of 23.44° with the plane of its orbit around the Sun, and the eccentricity of the Earth's orbit, which, in accordance with Kepler's second law, imposes variations in the planet's orbital velocity. The morphology of the analemma is contingent on the observer’s latitude and the specificity of the measurement instant, manifesting itself in its most recurrent conception as an elongated curve of elegiac configuration.

An analemma can be described mathematically using the equation of time and the variation of solar declination, as described by Pereira et al. (2025).

The equation of time, expressed as follows, demonstrates the difference between apparent solar time (based on the actual position of the Sun) and mean solar time (based on a fictitious Sun moving at a constant speed throughout the year):

$E_t=9.87\sin \left(2B\right)-7.53\cos \left(B\right)-1.5\sin \left(B\right)$

Where,

$B=\frac{360}{365}(N-81)$

N is the day number in the year (1 for January 1, and 365 for December 31).

Et is given in minutes and can be positive or negative.

The solar declination represents the latitude at which the sun's rays are perpendicular to the Earth's surface and varies throughout the year according to the tilt of Earth's axis.

$\delta =23,44\sin \left[\left(\frac{360}{365}\right)\right(N+10\left)\right]$

N is the day number in the year (1 for January 1, and 365 for December 31).

The equation of time affects the longitude of the Sun (the difference between apparent and mean solar time). Solar declination, in contrast, affects the latitude of the Sun (height in the sky). When these two quantities are plotted over the course of a year, the classic figure of the analemma in the shape of an octagon is obtained, as shown in Figures 1 and 2.

Figure 1
Movement of the solar analemma by geodesics throughout the year on the globe (available on the SunEarthTools.com platform, 2025).

Figure 2
Elevation of the movement of the solar analemma by geodesics throughout the year on the globe (available on the SunEarthTools.com platform, 2025).

Results and Discussion

From the daily incident solar radiation data on surfaces inclined at 10° and 20°, and considering the four cardinal orientations (north, south, east, and west) and the four seasons of the year (summer, autumn, winter, and spring), it was possible to generate graphs representing the global solar radiation throughout the day for each configuration. Based on this data, additional graphs were prepared that synthesized energy losses over time by comparing the efficiency of solar capture in different orientations and identifying the direction of greatest potential for energy generation, which was found to face north. To understand the dynamics of the energy generated throughout the year, dispersion plots representing the differential of energy generated over time were constructed.

Surprisingly, this graph presented a geometric pattern characteristic of a solar analemma (Figures 3 and 4), a curve that describes the apparent position of the Sun in the sky throughout the year, resulting from the combination of the Earth's axial tilt and the variation in orbital velocity due to the eccentricity of the Earth's orbit. This discovery led to the hypothesis that the energy generated by solar systems can be analyzed not only as a scalar quantity, but also as a vector or even a tensor, whose components vary as a function of inclination, orientation, and time of year. Such an approach allows the distribution of energy in a curved space to be described, the structure of which can be analyzed using differential geometry. Thus, a new paradigm for solar energy modeling is proposed, using advanced mathematical tools to understand and optimize its capture.

Figure 3
The analemmatic pattern in the difference in energy generated throughout the year.

Figure 4
The analemmatic pattern in the difference in energy generation throughout the year.

Riemannian geometry and the analemma

The analemma can be described as a parametric curve in a Riemannian manifold, where we consider a curved space–time, modeled by a metric tensor guv. We can represent the apparent displacement of the Sun in the sky as a curve embedded in a pseudo-Riemannian manifold with the following metric:

$d{s}^2=g_{\mu v}d{x}^{\mu }d{x}^v$

where the Sun's trajectory (the analemma) can be treated as a modified geodesic owing to the equation of time and solar declination. Therefore, the analemma can be treated mathematically as a parameterized curve in a Riemannian manifold. It represents the apparent trajectory of the Sun in the sky throughout the year in a spherical coordinate system (t,θ,φ), where solar declination (θ) and the equation of time (φ) can be described by periodic functions. To model it, an appropriate metric was defined and the differential equation that describes its evolution was found.

Riemannian space for the analemma

The space in which the apparent motion of the Sun occurs can be represented as a Riemannian manifold, M, equipped with the metric tensor guv. We consider the analemma as a curve, γ(λ), parameterized by an affine parameter λ, which runs along a 2D surface embedded in a 4D pseudo-Riemannian manifold of space–time. If we consider the projection of the Sun's trajectory in the sky (analemma) as a curve on a spherical surface, the natural metric for this space is the metric of the unit sphere.

$d{s}^2=d{\theta }^2+{\sin }^2\theta d{\varphi }^2$

Where:

t is the observer's own time;

θ is the solar declination δ;

φ is the apparent longitude corrected using the equation of time.

However, to incorporate the dynamics of the Earth's orbit and its eccentricity, we can modify this metric for a deformed space by considering a time-dependent factor.

$d{s}^2=d{\theta }^2+f(\theta ,t)d{\varphi }^2$

Where:

f(0, t) represents the change in the angular position of the Sun owing to the equation of time.

Curvature and equation of the Riemann tensor for the analemma

The geometry of the surface on which the analemma is embedded is described by the Riemann curvature tensor as follows:

$R_{\sigma \mu v}^{\rho }={\partial }_{\mu }{\Gamma }_{v\rho }^{\sigma }-{\partial }_v{\Gamma }_{\mu \rho }^{\sigma }+{\Gamma }_{\lambda \mu }^{\rho }{\Gamma }_{\sigma v}^{\lambda }-{\Gamma }_{\lambda v}^{\rho }{\Gamma }_{\sigma v}^{\lambda }$

Where:

Christoffel (𝛤^𝜌_μ𝜐) symbols are defined by the following metric:

${\Gamma }_{\mu v}^{\lambda }=\frac{1}{2}{g}^{\lambda \sigma }({\partial }_{\mu }{g}_{\sigma v}+{\partial }_v{g}_{\sigma }-{\partial }_{\sigma }{g}_{\mu v})$

For a two-dimensional space with the metric of type $d{s}^2=d{\theta }^2+f(\theta ,t)d{\varphi }^2$, the only relevant term of the Ricci scalar curvature is:

$R=-\frac{1}{f}\frac{d^{2}f}{d{\theta }^2}$

Differential equation for the analemma

We can describe the analemma as a parametric curve (γ(λ)) in the manifold M, where the geodesic equation is given by:

$\frac{d^2{x}^{\mu }}{d{\lambda }^2}+{\Gamma }_{\rho \sigma }^{\mu }\frac{d{x}^{\rho }}{d\lambda }\frac{d{x}^{\sigma }}{d\lambda }=0$

Substituting the spherical coordinates x^μ= (θ,φ), we obtain two differential equations for solar declination and longitude, as follows:

• For solar declination θ:

$\frac{d^2{x}^{\mu }}{d{\lambda }^2}+{\Gamma }_{\theta \theta }^{\theta }{\left(\frac{d\theta }{d\lambda }\right)}^2+{\Gamma }_{\varphi \varphi }^{\theta }{\left(\frac{d\varphi }{d\lambda }\right)}^2=0$

• For solar longitude φ:

$\frac{d^2\varphi }{d{\lambda }^2}+2{\Gamma }_{\theta \varphi }^{\varphi }\frac{d\theta }{d\lambda }\frac{d\varphi }{d\lambda }=0$

By expanding these terms with the Christoffel symbols calculated for the spherical metric, we obtain the governing differential equation for the analemma.

Approximate solution and relationship to differential geometry

The approximate parametric solution of the analemma can be expressed as:

$\theta \left(\lambda \right)=23,45\sin \left[\left(\frac{2\pi }{365}\right)\right(\lambda +10\left)\right]$

$\theta \left(\lambda \right)=9,87\sin \left[\left(\frac{4\pi }{365}\right)\right(\lambda -81\left)\right]-7,53\cos \left[\left(\frac{2\pi }{365}\right)\right(\lambda -81\left)\right]-1,5\sin \left[\left(\frac{2\pi }{365}\right)\right(\lambda -81\left)\right]$

By inserting these expressions into the equation of the scalar curvature R, we can understand how the surface on which the analemma is embedded curves in space–time. To consider the variation in the Earth's orbit and its eccentricity, we can define f(θ,t) based on the variations in the equation of time. As a first approximation,

$f(\theta ,t)={\sin }^2\theta +ϵg(\theta ,t)$

where 𝝐𝒈(𝜽,𝒕) represents perturbations due to the eccentricity of the Earth's orbit and tilt of the axis. By deriving f(θ,t) twice with respect to θ, we obtain the scalar curvature R as follows:

$\frac{df}{d\theta }=2\sin \theta \cos \theta +ϵ\frac{dg}{d\theta }$

$\frac{d^{2}f}{d{\theta }^2}=2{\cos }^2\theta -2{\sin }^2\theta +ϵ\frac{d^{2}g}{d{\theta }^2}$

$=2\cos 2\theta +ϵ\frac{d^{2}g}{d{\theta }^2}$

By inserting the above into the equation of scalar curvature, the general global equation of the curvature of space is obtained, in which the analemma is embedded and its variation with time and solar declination is shown. The eccentricity of the orbit and the Earth’s axial tilt (through the equation of time) deform the geometric surface, thereby influencing the curvature R.

$R=-\frac{1}{{\sin }^2\theta +ϵg}(2\cos 2\theta +ϵ\frac{d^{2}g}{d{\theta }^2})$

Energy as a tensor field

Energy can be described by an energy flow tensor, which in general relativity relates to the curvature of space–time through Einstein's equation:

$G^{\mu v}=\frac{8\pi G}{c^4}{T}^{\mu v}$

where 𝑇^𝜇𝜐 is the Einstein tensor (derived from the curvature of, and describes the distribution of energy and momentum. In this study, we considered simplification as a hypothesis: the energy generated throughout the year can be described as a scalar field 𝐺^𝜇𝜐𝑅_𝜇𝜐𝑇^𝜇𝜐E(λ,θ,φ)E(λ,θ,φ), where λ represents the time throughout the year, θ the slope, and φ the orientation.

We define an energy flow tensor such that, on a two-dimensional surface (𝑇^𝜇𝜐θ,φ), it can be expressed in a simplified form:

$T^{\mu v}=\sigma E{g}^{\mu v}$

where 𝒈^𝝁𝝊 is the surface metric, and σ is a proportionality factor that can depend on the slope. The fundamental equation that relates energy to scalar curvature ® comes from the conservation of the energy tensor:

${\nabla }_{\mu }{T}^{\mu v}=0$

On explicit expansion:

${\nabla }_{\mu }\left(\sigma E{g}^{\mu v}\right)=0$

Or,

$\sigma {\nabla }_{\mu }\left(E{g}^{\mu v}\right)+E{g}^{\mu v}{\nabla }_{\mu }\sigma =0$

Because we are considering a curved surface (f(θ,t)) we must express the Laplacian in the analemma metric. The idea is to express the variation of the generated energy using a Laplace–Beltrami operator on the curved surface.

$\Delta E=RE$

where Δ is the Laplacian operator defined by the metric you used for the analemma, and R is the scalar curvature found earlier.

This implies that the energy varies according to the curvature of the surface of the analemma, which, in turn, can possess properties similar to those of a Möbius strip or Klein bottle, depending on the topology of the orientation of the solar panels. Consequently, the expression E(λ,θ,φ) will be deduced more formally. The Laplace–Beltrami operator for the metric of the analemma is calculated, and it is verified that it behaves as expected on a Möbius/Klein surface. Further, it is tested whether the energy follows a periodicity with an orientation transition (as on non-orientable surfaces).

Laplace–Beltrami operator for the analemma

The Laplacian in a Riemannian manifold is given by the divergence of the gradient, that is, in the analemma metric, we assume a metric such as:

$\Delta f={\nabla }^{\mu }{\nabla }_{\mu }f$

Considering the metric of the unit sphere, demonstrated mathematically above, the Laplace–Beltrami operator in this metric is expressed as follows:

$\Delta E=\frac{1}{\sqrt{g}}{\partial }_{\mu }\left(\sqrt{g}{g}^{\mu v}{\partial }_{v}E\right)$

For the given metric,

$g_{\mu v}=\left[\begin{array}{cc}1& 0\\ 0& f(\theta ,t)\end{array}\right]$

where g is the determinant of the metric, which can be calculated as follows:

$g=f(\theta ,t)$

Therefore, the inverse of the metric is:

$g^{\mu v}=\left[\begin{array}{cc}1& 0\\ 0& f(\theta ,t{)}^{-1}\end{array}\right]$

Expanding on the expression of the Laplacian:

$\Delta E=\frac{1}{\sqrt{f(\theta ,t)}}[{\partial }_{\theta }\left(\sqrt{f(\theta ,t)}{\partial }_{\theta }E\right)+{\partial }_{\varphi }\left(\frac{1}{\sqrt{f(\theta ,t)}}{\partial }_{\varphi }E\right)]$

$\Delta E=\frac{1}{\sqrt{f(\theta ,t)}}\frac{{\partial }^{2}E}{{\partial }^2\theta }+\frac{1}{f(\theta ,t)}\frac{d}{d\theta }\left(f\right(\theta ,t\left)\frac{dE}{d\theta }\right)+\frac{1}{f(\theta ,t)}\frac{{\partial }^{2}E}{\partial {\varphi }^2}$

We relate this equation to the scalar curvature R, found earlier, as follows:

$\frac{1}{f}\frac{{\partial }^{2}E}{{\partial }^2\theta }+\frac{1}{f}\frac{d}{d\theta }\left(f\frac{dE}{d\theta }\right)+\frac{1}{f}\frac{{\partial }^{2}E}{\partial {\varphi }^2}=-RE$

We can describe the variation of the energy generated throughout the year as a differential equation coupled to the curvature of the analemma surface:

$\Delta E+RE=0$

This Laplacian appears in several differential equations, such as the diffusion or Helmholtz equations, when applied to the deformed surface of the analemma. It can be used to model the variation of the energy generated over time and seasons.

The evolution of the generated energy can be modeled using a modified Helmholtz-type equation as follows:

$\Delta E+R\left(\theta \right)E=S(\theta ,\varphi ,t)$

Where:

ΔE is the Laplacian of E(θ,φ) in the analemma metric;

R(θ) is the scalar curvature calculated earlier;

S(θ,φ,t) represents an external source (variation in solar irradiance throughout the year).

Substituting the above in the Laplacian above, together with the Ricci scalar curvature, we have

$\frac{1}{f}\frac{{\partial }^{2}E}{{\partial }^2\theta }+\frac{1}{f}\frac{d}{d\theta }\left(f\frac{dE}{d\theta }\right)+\frac{1}{f}\frac{{\partial }^{2}E}{\partial {\varphi }^2}-\frac{1}{f}\frac{d^{2}f}{d{\theta }^2}E=S(\theta ,\varphi ,t)$

The relationship between the analemma and differential geometry can be explored through the theory of curved surfaces and Riemann geometry, and its shape follows the pattern found in the Mobius ribbon and Klein bottle. Both are non-orientable surfaces, and may have an interesting connection with the study: throughout the year, depending on the inclination and orientation of the solar panels, the energy generated may present patterns that do not close in a conventional way, but rather with a topological twist. However, stating that the analemma is a Möbius strip is not rigorously correct, as the Möbius strip is a non-orientable surface, whereas the analemma, being a curve drawn in space–time (assuming a spherical coordinate system), follows an orientable topology.

This approach reinforces the hypothesis that the variation in the energy generated throughout the year can be described in a space with non-trivial topological properties. The presence of the Möbius strip or Klein bottle as underlying structures suggests that the evolution of energy over time involves a continuous transition, but with a twist that prevents a well-defined global orientation. For the Klein bottle, it will be necessary to extend this formulation to a space with coordinates (u,v,w)(u, v, w)(u,v,w), introducing additional terms that capture the connectivity of the surface in space–time, although in this article we will not address such a methodology because the same equation would be close to the Möbius strip. Moreover, it is suggested for future work to study the topology in the Klein bottle, including a 4D parameterization.

On the Möbius strip, we can parameterize the surface by coordinates (u,v)(u,v)(u,v), where

u runs the length of the ribbon (temporal direction throughout the year, equivalent to λ);

V runs across the width of the tape and is cyclically identified with a signal inversion.

This redesign allows us to explore how geometry influences energy variation and provide insights into optimizations based on the topological structure of solar capture. The parameterization is as follows:

$x=(1+v\cos \left(\frac{u}{2}\right))\cos u$

$y=(1+v\cos \left(\frac{u}{2}\right))\sin u$

$z=v\sin \left(\frac{u}{2}\right)$

Where u ∈ [0,2π] and v ∈ [−1,1].

The relationship with the generated energy can be written as follows:

$\frac{d^{2}E}{d{\lambda }^2}+f(\theta ,\varphi ,\lambda )\frac{dE}{d\lambda }+RE=0$

By exchanging λ for the intrinsic coordinate u and considering that the differential operator must respect the structure of the surface, we use the local metric gik of the Möbius strip, which is the induced metric of the parameterized surface with its metric in the coordinates you and v:

$d{s}^2={(1+v\cos \left(\frac{u}{2}\right))}^{2}d{u}^2+d{v}^2$

The metric determinant is:

$g={(1+v\cos \left(\frac{u}{2}\right))}^2$

The Laplace–Beltrami operator, demonstrated earlier, on the surface of the Möbius strip can then be expressed by substituting it into the differential energy equation proposed in the present study. Thus, we obtain

$\frac{1}{(f1+v\cos \left(\frac{u}{2}\right))}\frac{\partial }{\partial u}\left((1+v\cos \left(\frac{u}{2}\right))\frac{\partial E}{\partial u}\right)+\frac{{\partial }^{2}E}{\partial v^2}+f(u,v)\frac{\partial E}{\partial u}+RE=0$

New general differential equation for the exergy of PV systems

Exergy measures the useful portion of the energy available for conversion into work, considering system irreversibilities such as seasonal variations and changes in efficiency arising from the tilt and orientation of the panels.

The exergy equation in thermal systems can be written as:

$\frac{dX}{dt}=Q(1-\frac{T_0}{T})-X_{destr}$

Where:

X is the exergy of the system;

Q is the incident heat flux;

T0 is room temperature;

T is the temperature of the system;

Xdestr is the destroyed exergy due to irreversibility.

We can establish a direct association between the equations if we assume that the energy generated is proportional to the exergy X of the solar system and identify the corresponding terms.

• The term scalar curvature RE can represent the destruction rate of exergy X_dexter, which depends on atmospheric conditions and thermal losses;

• The external term F(θ,φ,λ) can be associated with the exergy available in the solar system, equivalent to $Q(1-\frac{T_0}{T})$;

• The Laplace–Beltrami operator may represent the diffusion of exergy owing to seasonal changes and variations in orientation.Δ𝐸

Thus, we arrive at a new differential equation associated with the exergy of PV systems as follows:

$\Delta X+RX=Q(1-\frac{T_0}{T})$

This equation indicates that the distribution of exergy throughout the year is not homogeneous and is influenced by the curvature of the space–time of the analemma, reflecting the geometric variation of the energy available for conversion. This demonstrates that the analysis of exergy can be completely integrated into its study, reinforcing the interpretation that the variation in the generated energy follows a geometric behavior similar to that of the Möbius ribbon and Klein bottle.

Discussion

This work proposes an innovative approach by treating the variation in solar energy generated throughout the year as a phenomenon governed by non-trivial geometric and topological structures modeled in a Riemannian space with properties analogous to the Möbius ribbon and Klein bottle. From the empirical analysis of solar radiation data on surfaces inclined at 10° and 20°, considering the four cardinal orientations (north, south, east, and west) and the four seasons of the year, a scattering pattern was identified that reproduced the shape of the solar analemma, a figure that describes the apparent position of the Sun in the sky throughout the year, as a function of the equation of time and solar declination.

Avila & Raza (2024) focused on the mathematical and computational description of the solar trajectory, using rotation methods in Cartesian systems and vector algorithms to calculate the position of the Sun. Their proposal offers a robust tool for solar engineers based on classical Euclidean geometry. Although both studies share practical interest in the representation of the analemma, this work advances their findings by proposing a differentiated geometric interpretation, treating the generated energy as a tensor field, whose behavior is conditioned by the curvature of space, in analogy to the field approach of general relativity.

When comparing classical and differential models, the need for geometric complexification is justified: although there are effective methods for calculating the solar position (Avila & Raza, 2024), there is still a gap in the use of differential geometry and topology as interpretative tools for energy variation. Our study transcends merely celestial periodicity and proposes a model that incorporates scalar curvature, non-orientability, and parallel energy transport in dynamic topological manifolds.

In addition, the analysis conducted by Zhang et al. (2021a) reinforces the vector basis of the solar trajectory, modeling it from a three-dimensional unit vector S⃗\vec{S}S and evidencing the periodicity of solar positions in an "analema corona.” Although based on Cartesian geometry, their findings provide support for the graphical visualization that inspires the extension proposed herein: by considering the energy variation as dependent on the scalar curvature of the analemma metric and applying the Laplace–Beltrami operator, we evidence that the generated energy manifests complex topological patterns, close to non-orientable structures such as the Möbius ribbon and Klein bottle.

These observations are corroborated by the three-dimensional representations of the analemma presented by Avila & Raza (2024) and Zhang et al. (2021b), in which a saddle curvature is perceived—a characteristic of hyperbolic surfaces typical of Riemannian geometry—when observing the analemma in a lateralized manner. Such behavior reinforces the hypothesis that the structure of the analemma and the dynamics of solar energy are intrinsically linked to the topological and differential properties of energetic space–time.

Finally, consistent with Vasilyev et al. (2013) theoretical proposal, which views non-orientable surfaces as fundamental principles of natural organization, this work proposes that the difference in the generated solar energy can be described as a tensor field over a non-trivial topological manifold. The integration of exergy analysis in this context reveals that the quality of the energy available throughout the year is profoundly influenced by the geometric structure in which it manifests. Thus, the solar analemma is reinterpreted not only as an astronomical artifact, but as a direct expression of a curved and non-orientable energy field, the geometric understanding of which affords new perspectives for the optimization of solar systems.

The current study seeks to expand the understanding of the relationship between the variation in solar energy generated throughout the year and the differentiated geometry of non-orientable topological surfaces, such as the Möbius strip and Klein bottle. The analysis of the incident solar radiation data on surfaces inclined at 10° and 20°, considering different cardinal orientations and seasonal variations, revealed a peculiar behavior of the dispersion of the values of the differential generated energy, which presents a structure analogous to a solar analemma. This finding suggests that the behavior of solar energy can be treated from the perspective of a geometric tensor system, in which the seasonal and angular variations of energy capture can be modeled in a space with non-trivial properties.

The connection between solar energy and non-orientable surfaces is supported by the principles addressed by Cimasoni et al. (2013), who proposed a unification of physical and mathematical phenomena based on the concept of torsion and the logic of the Klein bottle and Möbius strip. His work highlights the importance of non-orientable surfaces in describing nonlinear systems, resonances, and wavefront propagation in diverse fields, from cosmology to biology. This theoretical framework justifies the hypothesis that the variation in solar energy throughout the year can be modeled in a more robust manner through differential geometric theory, which allows the description of transitions of orientability and the nonlinear effects present in the spatial and temporal distribution of the captured energy.

By inserting the metric of the analemma in the formulation of the Laplacian and relating it to the scalar curvature, it was possible to mathematically describe the distribution of solar energy in a space with a differentiated structure. The hypothesis that the generated energy can be treated as a tensor, with explicit dependence on the scalar curvature of the surface formed by the data over time, finds a parallel in Cimasoni et al. (2013) approach to vector fields and their applications in geophysical physics and pattern recognition. Thus, the methodology adopted in this study not only reinforces the validity of the initial hypothesis, but also proposes a new perspective on the modeling of solar energy on surfaces that challenge traditional conceptions of orientability and spatial continuity.

In addition, the concept of exergy proves to be fundamental in this analysis, as it allows the quantification not only of the available energy, but also its quality and use in different orientations and inclinations of the solar panels. The connection between the theory of exergy and the tensor analysis of the generated energy reinforces the applicability of this study to energy optimization systems and the development of more accurate models for the use of solar radiation.

The article entitled "Hipparchos and the Ancient Analemma," by Nathan Sidoli (2024), aligns significantly with this work by investigating the analemma as a geometric representation of complex astronomical phenomena—and this convergence occurs both in historical and mathematical perspectives. Sidoli shows how Hipparchus, using ancient geometrical methods such as analemma diagrams and trigonometry with string tables, was able to solve fundamental astronomical problems, such as calculating right ascension and ecliptic position with high precision, employing a rigorous analytical approach based on spherical geometry and projections.

This approach finds a direct echo in his work, which proposes a contemporary and differential interpretation of the analemma to model the variation of solar energy throughout the year. Just as Hipparchus used the analemma to convert astronomical coordinates and predict celestial positions, the analemma can be used as a basis for describing the temporal and spatial distribution of solar energy, including its twisted and nonlinear patterns, which evoke surfaces such as the Möbius ribbon and Klein bottle. Although separated by millennia and specific objectives, both studies used the analemma as a geometric and heuristic tool to understand dynamical systems governed by natural cycles and non-trivial projections.

In addition, the concept of "lines" (διὰ τῶν γραμμῶν), which in Sidoli's article represents the use of geometric constructions to describe celestial phenomena, finds a parallel in his proposal to associate the differential of solar energy to a vector field on a Riemannian manifold, a field that is also constructed "by means of lines," only now in the modern sense, with tensors, scalar curvature, and dynamic metrics.

Therefore, Sidoli's work can be cited as an epistemological and historical basis, justifying that the use of the analemma as a computational and geometric model has a consolidated tradition and that its extrapolation to the field of contemporary solar energy is not only legitimate, but coherent with the evolution of astronomical–mathematical thinking since antiquity.

Thus, this work not only proposes a new perspective for the analysis of solar energy, but also dialogues directly with contemporary approaches that use non-orientable topological surfaces to describe complex systems. The study aligns with Cimasoni et al. (2013) work by demonstrating how concepts of torsion, vector fields, and differentiable surfaces can be applied to understand physical phenomena in a more comprehensive way, breaking with dualistic paradigms and providing new interpretations for energy variation throughout the year.

Conclusions

The analemma can be described in the formalism of Riemannian geometry as a geodesic curve on a curved manifold, whose Riemann tensor defines its local geometry. However, the analemma itself is not a Möbius strip, as it is a closed, steerable curve without an intrinsic topological inversion. With a three-dimensional parameterization, we could add a mathematical twist to construct a structure similar to a Möbius strip; however, this does not occur naturally in astronomical contexts.

The developed analysis revealed an inverse relationship between the global effective incident energy and the generated energy, considering different inclinations and orientations throughout the seasons, resulting in a scatter plot whose morphology resembles a solar analemma. In addition, by incorporating the influence of the inclination and orientation of the PV panels, this structure assumes geometric characteristics that evoke a Möbius ribbon and Klein bottle, suggesting an intrinsic relationship between the space–time curvature and the seasonal variation in energy capture.

The formulation of the scalar curvature of the system, combined with the application of the Laplace–Beltrami operator in the analemma metric, allowed us to model the distribution of energy throughout the annual cycle, evidencing a correlation between the geometric properties of space and the dynamics of energy capture. Thus, the obtained results reinforce the possibility of a more in-depth mathematical treatment of the interaction between differential geometry and the optimization of energy efficiency, opening new perspectives for the advanced modeling of solar capture systems.

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• Data Availability Statement:
  The datasets generated and/or analyzed during the current study are available from the corresponding author upon reasonable request.
• Funding:
  This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – CAPES (Brazilian Federal Agency for Support and Evaluation of Graduate Education) – Finance Code 001.

Edited by

• Area Editor:
  Juliana Lobo Paes

Data availability

The datasets generated and/or analyzed during the current study are available from the corresponding author upon reasonable request.

Publication Dates

• Publication in this collection
  27 July 2026
• Date of issue
  2026

History

• Received
  7 July 2025
• Accepted
  3 Apr 2026