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April 13, 2026Open Access

The Riemannian Geometry of the Prime Gravity Manifold: ArcLength Isometry and the Geodesic Schrödinger Operator

Key Points

To establish the Riemannian geometry of the Prime Gravity manifold and prove an arc-length isometry theorem.
Developed geometric framework for the Prime Gravity manifold
Proved isometry theorem involving the Laplace-Beltrami operator
Analyzed the Schrödinger operator in arc-length coordinates using Sturm-Liouville theory
Demonstrated the exact simplification of the Laplace-Beltrami operator to a flat Schrödinger operator
Showed that the operator is self-adjoint on L² space
Provided a foundation for the Hilbert–Pólya investigation based on geometric principles

Abstract

We develop the Riemannian geometry of the Prime Gravity manifold and prove an exact arc-length isometry theorem: the Laplace-Beltrami operator on the manifold with metric g₁₁ = 1 + (dVPG/du) ² simplifies exactly under the arc-length coordinate s (u) = ∫√g dt to the flat Schrödinger operator Hgeo = −d²/ds² + VPG (u (s) ), with no approximation and no residual first-derivative terms. The operator is self-adjoint on L² (0, sₘax, ds) via standard Sturm-Liouville theory. This provides the geometric and operator-theoretic foundation for the Hilbert–Pólya investigation of Paper 8.

The Riemannian Geometry of the Prime Gravity Manifold: ArcLength Isometry and the Geodesic Schrödinger Operator

Key Points

Abstract

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