PreprintarXiv licencearXiv
Reconstruction of Heat Geometry from a Scalar Process
Abstract
A smooth scalar process with one fixed Gaussian covariance kernel determines any finite graded family of symmetric Hilbert tensors up to a common orthogonal isomorphism of their minimal supports. For two heat tensors, its law determines a closed positive-dimensional Riemannian manifold, a Euclidean bundle, its connection and potential, and the unknown heat time, up to isometry and gauge. The quadratic tensor recovers the scaled generator; the quartic tensor recovers the spatial norm and volume, which fix the time scale. For compact finite-dimensional RCD spaces, the same law determines the measured space with distance divided by the square root of the heat time.
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BibTeX
@misc{assaad2026reconstruction,
title = {{Reconstruction of Heat Geometry from a Scalar Process}},
author = {Obayda Julien Assaad},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.27967},
eprint = {2609.27967},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27967},
}
RIS
TY - UNPB
TI - Reconstruction of Heat Geometry from a Scalar Process
AU - Obayda Julien Assaad
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2609.27967
UR - https://arxiv.org/abs/2609.27967
AB - A smooth scalar process with one fixed Gaussian covariance kernel determines any finite graded family of symmetric Hilbert tensors up to a common orthogonal isomorphism of their minimal supports. For two heat tensors, its law determines a closed positive-dimensional Riemannian manifold, a Euclidean bundle, its connection and potential, and the unknown heat time, up to isometry and gauge. The quadratic tensor recovers the scaled generator; the quartic tensor recovers the spatial norm and volume, which fix the time scale. For compact finite-dimensional RCD spaces, the same law determines the measured space with distance divided by the square root of the heat time.
ER -