Tensor-Bundle Parametrization of Diffusion Densities

Definitions and mathematical statement

Let $(M,g)$ be the Riemannian manifold appearing in the source. Define the bundle of symmetric positive-definite covariant tensors of order $2$ by \[ \mathcal P(M)=\bigsqcup_{x\in M}\mathcal P_x(M),\qquad \mathcal P_x(M)=\left\{A\in\operatorname{Sym}^2(T_x^*M): A(v,v)>0\text{ for every }v\in T_xM\setminus\{0\}\right\}. \] Thus each $A\in\mathcal P_x(M)$ is a symmetric positive-definite bilinear form on $T_xM$.

Let $\Gamma$ denote the family described in the supplied record as the point-sourced diffusion densities used in diffusion principal component analysis on $M$.

Source conjecture. Prove or disprove that $\Gamma$ can be parametrized by $\mathcal P(M)$, through a correspondence of the form \[ \Phi:\mathcal P(M)\longrightarrow\Gamma. \] Unresolved qualifications. The supplied record does not define the diffusion construction or $\Gamma$ sufficiently to specify $\Phi$. It does not specify the observation time, the reference measure for densities, the admissible manifold hypotheses, or boundary and lifetime conventions. It also does not specify whether parametrization means a surjection, a bijection, a homeomorphism, or a smooth identification, nor which topology or smooth structure is intended on $\Gamma$. Whether $\Phi(A)$, for $A\in\mathcal P_x(M)$, must have source point $x$ is likewise unstated. Consequently, this is the source conjecture with its missing specifications exposed; the supplied evidence does not determine a fully quantified assertion for a particular diffusion family.

Short English statement

The conjecture asks whether the point-sourced diffusion densities used in diffusion PCA on a Riemannian manifold can be parametrized by positive-definite symmetric covariant $2$-tensors, with their base points included. The diffusion family and the required meaning of parametrization remain unspecified in the supplied record.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002681$, title Tensor-Bundle Parametrization of Diffusion Densities, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The record lists proposed year 2014 and supplies no proposer attribution.
• Supplied report citation: Mini-Workshop: Asymptotic Statistics on Stratified Spaces (2015), Oberwolfach Reports, pp. 2481--2527, DOI 10.4171/owr/2014/44. https://doi.org/10.4171/owr/2014/44. The literature-source list labels this reference Oberwolfach Report 44/2014. The supplied 2015 workshop designation and 2014 report designation require reconciliation.
• Supplied literature reference: Sommer, Anisotropic Distributions. https://doi.org/10.1007/978-3-319-19992-4_15. No further bibliographic details are supplied.
• Literature assessment checked at 2026-08-21: the supplied dated curation labels the problem open, reports subsequent work on anisotropic distributions and most-probable paths, and describes the topology and geometric parametrization of $\Gamma$ by positive-definite covariant tensors as unresolved. This is supplied dated curation, not independently verified literature assessment; its status and cited claims require independent verification.
• The registry supplies statement status corrected_verified, with an editorial note that mathematical content was preserved while notation, typography, or omitted definitions were repaired for publication. These are supplied metadata claims, not independent verification of this draft or its mathematical completeness.