Frame eversion and contextual geometric rigidity

TL;DR

This paper establishes rigidity results for maps on Grassmannians and manifolds of line tuples in finite-dimensional and Hilbert spaces, classifying structure-preserving transformations and revealing new symmetries.

Contribution

It provides new classification theorems for maps preserving geometric and lattice structures in Grassmannians and line manifolds, including novel contextual symmetries.

Findings

Maps preserving dimension and lattice operations are induced by semilinear injections.

Symmetric maps respecting linkage are characterized by semilinear injections or new contextual symmetries.

Identifies a new type of purely-contextual-global symmetry transforming line tuples.

Abstract

We prove rigidity results describing contextually-constrained maps defined on Grassmannians and manifolds of ordered independent line tuples in finite-dimensional vector or Hilbert spaces. One statement in the spirit of the Fundamental Theorem of Projective Geometry classifies maps between full Grassmannians of two $n$-dimensional Hilbert spaces, $n\ge 3$, preserving dimension and lattice operations for pairs with commuting orthogonal projections, as precisely those induced by semilinear injections unique up to scaling. In a different but related direction, denote the manifolds of ordered orthogonal (linearly-independent) $n$-tuples of lines in an $n$-dimensional Hilbert space $V$ by $\mathbb{F}^{\perp}(V)$ (respectively $\mathbb{F}(V)$) and, for partitions $\pi$ of the set $\{1..n\}$, call two tuples $\pi$-linked if the spans along $\pi$-blocks agree. A Wigner-style rigidity theorem proves that the symmetric maps $\mathbb{F}^{\perp}(\mathbb{C}^n)\to \mathbb{F}(\mathbb{C}^n)$, $n\ge 3$ respecting $\pi$-linkage are precisely those induced by semilinear injections, hence by linear or conjugate-linear maps if also assumed measurable. On the other hand, in the $\mathbb{F}(\mathbb{C}^n)$-defined analogue the only other possibility is a qualitatively new type of purely-contextual-global symmetry transforming a tuple $(\ell_i)_i$ of lines into $\left(\left(\bigoplus_{j\ne i}\ell_j\right)^{\perp}\right)_i$.