Frame eversion and contextual geometric rigidity

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 $\left(\left(\bigoplus_{j\ne i}\ell_j\right)^{\perp}\right)_i$.