Higher-Order Principal Components on Riemannian Manifolds

Definitions and mathematical statement

A Riemannian manifold $(M,g)$ is a smooth manifold equipped with a smoothly varying positive-definite inner product $g_p$ on each tangent space $T_pM$. In the usual smooth-distribution formulation, sub-Riemannian data consist of a distribution $\mathcal D\subseteq TM$ and a smoothly varying inner product on $\mathcal D$; admissible curves have tangent vectors in $\mathcal D$. The source does not specify such data or any bracket-generating hypothesis.

Conjectural target supplied by the source. Can higher-order derivative constructions of barycentric or affine subspaces, together with their associated principal-component analysis, be extended from spheres to arbitrary Riemannian manifolds by means of sub-Riemannian geometry? The supplied background also includes the associated unexplained-variance framework among the intended objects of extension.

Unresolved formulation. This is a research question rather than a fully specified mathematical proposition. The record supplies neither the spherical constructions to be extended nor definitions of their higher-order derivatives, barycentric or affine subspaces, principal components, and unexplained variance. It does not specify derivative orders, data or probability measures, optimization objectives, required extension properties, or the sub-Riemannian structure to be used. It also gives no dimension, completeness, curvature, boundary, or other hypotheses on the target manifolds. The phrase "arbitrary Riemannian manifolds" expresses the intended general scope, but the missing constructions and hypotheses prevent a precise quantified existence or equivalence statement. These qualifications remain unresolved in this statement-only draft.

Short English statement

The supplied conjectural question asks whether constructions of higher-order barycentric or affine subspaces on spheres, and their principal components and unexplained variance, extend to arbitrary Riemannian manifolds through sub-Riemannian geometry. The constructions and the properties an extension must preserve are unspecified, so a fully precise theorem statement requires further source clarification.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30003037$, title Higher-Order Principal Components on Riemannian Manifolds, source problem code catalogue entry, collection Oberwolfach Reports Open Problems. The supplied proposed year is 2016; no individual proposer is identified.
• Supplied report citation: Mathematical Imaging and Surface Processing (2016), Oberwolfach Reports, pp. 155--214, DOI 10.4171/owr/2016/4. Supplied source and literature reference, labeled Oberwolfach Report 4/2016: https://doi.org/10.4171/owr/2016/4. The record describes these reports as documenting mathematical workshops and their open-problem discussions; no pinpoint passage for this question is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that no general theorem validating the conjectured extension of the principal-component and unexplained-variance framework to higher derivatives on general manifolds was identified. This is supplied dated literature triage derived from the OWR curation, not an independently verified assessment. The record requests independent verification of time-sensitive status and cited claims.
• The supplied statement-status label is corrected_verified. Its accompanying verification note reports manual semantic, typographic, and LaTeX review and claims that the clean statement is self-contained and preserves the target without unsupported additions. These are supplied editorial assertions, not independent verification of the mathematical formulation or literature status.