Monistic conception of geometry

One considers the monistic conception of a geometry, where there is only one fundamental quantity (world function). All other geometrical quantities a derivative quantities (functions of the world function). The monisitc conception of a geometry is compared with pluralistic conceptions of a geometry, where there are several independent fundamental geometrical quantities. A generalization of a pluralistic conception of the proper Euclidean geometry appears to be inconsistent, if the generalized geometry is inhomogeneous. In particular, the Riemannian geometry appears to be inconsistent, in general, if it is obtained as a generalization of the pluralistic conception of the Euclidean geometry.

💡 Research Summary

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The paper proposes a “monistic” conception of geometry in which a single fundamental quantity – the world‑function σ(P,Q) (the squared distance between two points) – is taken as the only primitive. All other geometric objects – the metric tensor, tangent vectors, affine connection, curvature, etc. – are defined as derivatives or functional combinations of σ. This stands in contrast to the traditional “pluralistic” view, where several independent primitives such as distance, angle, metric, connection and curvature are postulated separately and linked by a set of axioms.

The author first shows how the Euclidean geometry can be reconstructed from the simple world‑function σ_E(P,Q)=½‖P−Q‖². By differentiating σ_E with respect to its arguments one recovers the Euclidean metric, the Levi‑Civita connection (which vanishes), and all familiar Euclidean constructions. The monistic approach thus reduces the entire Euclidean apparatus to a single scalar function.

The paper then attempts to generalize this scheme to inhomogeneous (non‑uniform) spaces. If σ is allowed to be an arbitrary smooth function of its two arguments, the derived metric g_{ij}(x)=∂i∂’jσ|{x’=x} and the derived connection Γ^k{ij}=½ g^{kl}(∂i g{jl}+∂j g{il}−∂l g{ij}) remain well‑defined provided σ is sufficiently differentiable. Consequently, a whole class of non‑Euclidean, possibly highly irregular geometries can be generated without introducing any new primitive.

In the pluralistic framework, however, distance d(P,Q) and metric g_{ij}(x) are introduced independently. The connection is then defined (for instance) as the unique torsion‑free, metric‑compatible Levi‑Civita connection of g. When the underlying space is inhomogeneous, the two independent specifications can become incompatible. The paper demonstrates that the “distance‑metric consistency condition” – namely that the metric must coincide with the second derivatives of the distance function – is not automatically satisfied when d and g are prescribed separately. Moreover, the “metric‑connection consistency condition” (∇k g{ij}=0) may fail if the connection is defined independently of the metric derived from σ.

A concrete illustration is given for the Riemannian case. Starting from a generic world‑function σ, one can compute a metric g_{ij}=∂_i∂’jσ. If one then imposes a different metric \tilde g{ij} (for example, one obtained from a separate physical requirement), the Levi‑Civita connection built from \tilde g will, in general, not coincide with the connection obtained by differentiating σ. Hence the usual Riemannian geometry, when viewed as a pluralistic extension of Euclidean geometry, becomes internally inconsistent for generic inhomogeneous manifolds. Only very special σ (e.g., those corresponding to spaces of constant curvature or possessing high symmetry) avoid this conflict.

To resolve the dilemma the author suggests two possible routes. The first is to retain the monistic stance but to restrict σ by additional smoothness or symmetry requirements, thereby limiting the class of admissible inhomogeneous geometries. The second is to adopt a “derived pluralism” in which all other primitives are explicitly required to be functions of σ; this guarantees internal consistency because the primitives are no longer independent.

The paper concludes that the monistic conception offers a dramatically simpler axiomatic foundation and a natural way to treat non‑uniform spaces, while the traditional pluralistic approach inevitably needs extra axioms or constraints to avoid contradictions. This insight has potential implications for fields that rely on geometric modeling of irregular spacetimes, such as general relativity, quantum gravity, and modern differential geometry.