Curvature Is Intrinsic: Riemann, and the Language for Not Conflating the Corpus’s Many “Curvatures”.
The headwater of the geometric-physics mainstream in which this corpus stands is Bernhard Riemann, and his revolution in one line is: curvature is INTRINSIC. The bending of a space is determined not by any embedding into an outer higher-dimensional space, but by the space’s own metric alone. Gauss’s Theorema Egregium (Gaussian curvature is invariant under isometry, fixed by the first fundamental form alone) was generalized by Riemann in 1854 to n dimensions, giving the Riemann curvature tensor built from the metric and its derivatives alone. Intrinsic means one can detect curvature from within (holonomy of parallel transport, geodesic deviation, the angle sum of a triangle) without stepping outside. This paper uses Riemann’s language as a tool to correctly sort the corpus’s several “curvatures”: spacetime curvature (general relativity), the constant velocity-space curvature K = -1/c^2 (Paper 60), and the sphere’s Gaussian curvature integrated to 4pi via Gauss-Bonnet (Papers 67/02). Riemann UNIFIES what curvature IS into one intrinsic concept, but that does not mean every space’s curvature is the same quantity: each is an intrinsic quantity of a different manifold, a distinct root merely sharing the word “curvature” (Papers 58/48). What is shared is the concept (intrinsicness); what must not be conflated is each instance. IMPORTANT (scope): a Layer-2 structural interpretation of standard Riemannian geometry; no new mathematics and no new prediction; Riemannian geometry is not extended. Riemann is the headwater of the corpus’s lineage (Riemann-Clifford-Einstein-Weyl-Cartan-Hopf-emergent gravity). “Matter = curvature” and the “Universe OS” are Layer-3 non-science lenses.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). Curvature Is Intrinsic [Paper 91]. Zenodo.
DOI: 10.5281/zenodo.21758406
ORCID: 0009-0006-8587-3450