Geodesics and intrinsic statistics of isolated connections in Hermitian positive-definite matrices
Abstract
Quantifying change in a single covariance, coherence or spectral connection requires a distance that respects the positive-definiteness constraint and holds the surrounding entries fixed. We characterize the Riemannian geometry induced by the affine-invariant metric when one conjugate off-diagonal pair of a Hermitian positive-definite matrix varies under this constraint. A fixed Schur congruence maps the admissible set to an open complex disk and reduces its metric to a two-dimensional form independent of the ambient matrix dimension. The disk is complete, with Gaussian curvature in (−1/4,0], and has unique minimizing geodesics and globally defined logarithms. Its intrinsic distance strictly exceeds the ambient affine-invariant distance between distinct points. Conservation of angular momentum reduces the geodesic boundary-value problem to a single scalar equation with elementary angular and length integrals, enabling distance and logarithm evaluation without shooting. For a declared reference distribution with a finite second intrinsic moment, we establish a unique Fréchet centre and consistency of its empirical estimate, together with a sandwich central limit theorem under the stated regularity conditions. Tangent covariance standardization yields moment identities, and exchangeable rank calibration supplies finite-sample score guarantees. These results provide a geometric and statistical framework for isolated connections on a fixed complement, with explicit limits on extension to varying complements.
Related
- Algebraic limits of Hermitian-Yang-Mills connections on Kähler surfaces
- Hermitian Connections with Parallel Torsion and the Fino--Vezzoni Conjecture
- Hermitian connections with parallel torsion and constant holomorphic sectional curvature
- Non-Hermitian Yang--Mills Connections with Vanishing Chern Classes
- Geodesics with Unified Tangent-constrained Priors and Curvature Regularization
BibTeX
@misc{reyes2026geodesics,
title = {{Geodesics and intrinsic statistics of isolated connections in Hermitian positive-definite matrices}},
author = {Ronald García Reyes and Pedro Antonio Valdés-Sosa},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01202},
eprint = {2611.01202},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01202},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Geodesics and intrinsic statistics of isolated connections in Hermitian positive-definite matrices
AU - Ronald García Reyes
AU - Pedro Antonio Valdés-Sosa
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01202
UR - https://arxiv.org/abs/2611.01202
AB - Quantifying change in a single covariance, coherence or spectral connection requires a distance that respects the positive-definiteness constraint and holds the surrounding entries fixed. We characterize the Riemannian geometry induced by the affine-invariant metric when one conjugate off-diagonal pair of a Hermitian positive-definite matrix varies under this constraint. A fixed Schur congruence maps the admissible set to an open complex disk and reduces its metric to a two-dimensional form independent of the ambient matrix dimension. The disk is complete, with Gaussian curvature in (−1/4,0], and has unique minimizing geodesics and globally defined logarithms. Its intrinsic distance strictly exceeds the ambient affine-invariant distance between distinct points. Conservation of angular momentum reduces the geodesic boundary-value problem to a single scalar equation with elementary angular and length integrals, enabling distance and logarithm evaluation without shooting. For a declared reference distribution with a finite second intrinsic moment, we establish a unique Fréchet centre and consistency of its empirical estimate, together with a sandwich central limit theorem under the stated regularity conditions. Tangent covariance standardization yields moment identities, and exchangeable rank calibration supplies finite-sample score guarantees. These results provide a geometric and statistical framework for isolated connections on a fixed complement, with explicit limits on extension to varying complements.
ER -