Form-Preserving Transformations

• Form-preserving transformations are mappings that maintain essential geometric, algebraic, or analytic properties across diverse mathematical structures.
• They are characterized by explicit analytic, group-theoretical, or algebraic conditions that enforce invariance under strict constraints.
• These transformations underpin canonical classifications in fields such as linear spaces, operator theory, PDE symmetry, and quantum dynamics, driving advances in solution techniques.

Form-preserving transformations are mappings between mathematical structures—linear spaces, operator fields, algebraic systems, geometric spaces, or solution spaces of differential equations—that preserve key geometric, algebraic, or analytic properties. These transformations maintain the "form" of the original objects, such as open segments, moments, structural matrix properties, or the solution set of PDEs, under explicit constraints. Rigorous classification of form-preserving transformations has been achieved in a variety of mathematical domains by exploiting functional equations, group-theoretical invariance, matrix theory, and analytic properties.

1. Segment- and Convexity-Preserving Maps on Linear Spaces

Given real linear spaces $X, Y$ and a convex domain $D \subset X$, a map $f: D \to Y$ is said to be segment-preserving if it takes open segments of $X$ to open segments in $Y$: for all $x,y \in D$, $$f(\langle x, y \rangle) = \langle f(x), f(y) \rangle$$, where $$\langle x, y \rangle := \{ tx + (1-t)y : 0 < t < 1\}$$ (Páles, 2012).

Explicit Characterization

If the image $f(D)$ is non-collinear (i.e., not contained in a single line), then every segment-preserving (form-preserving) transformation is rational of the form: $$f(x) = \frac{A(x) + a}{B(x) + b}, \qquad x \in D = \{ x \mid B(x) + b > 0 \}$$ where $A:X\to Y$ is linear, $a \in Y$, $B:X\to \mathbb{R}$ is linear, and $b \in \mathbb{R}$. The affine case ($B=0$) yields the familiar affine maps $f(x) = A(x) + a$.

In one dimension ($X = Y = \mathbb{R}$), the only form-preserving maps are strictly monotone (or constant) functions.

The non-collinearity criterion is essential: for collinear images, degenerate cases (monotonic maps) are possible.

Connection to Geometry and Quantum Structures

These results generalize classical projective geometry and are crucial, for example, in the structure theory of effect algebras in quantum measurement, where mixture-preserving automorphisms are forced to be affine (Páles, 2012).

2. Form-Preserving Transformations of Operator Fields: Toeplitz and Nijenhuis Operators

For operator fields $L$ of upper triangular Toeplitz form,

$$L = \begin{pmatrix} g_n & g_{n-1} & \cdots & g_1 \ 0 & g_n & \ddots & \vdots \ & & \ddots & g_{n-1} \ 0 & 0 & \cdots & g_n \end{pmatrix}$$

form-preserving coordinate transformations $v(u)$ are precisely those constructed as solutions to a recursive linear PDE system (Chernin et al., 14 Oct 2025): $M^* d v^i = d v^{i+1}, \quad i = 1,\ldots,n-1$ where $M$ is the associated Toeplitz Nijenhuis operator, and the transformation is parametrized by one function of one variable and $n-1$ functions of two variables. The classification is algorithmic—recursive integration and triangular linear algebra—yielding all coordinate changes that preserve the polynomial/Jordan block structure.

These characterizations generalize classic canonical form-preserving changes for diagonalizable operators to the non-diagonalizable (nilpotent/Jordan block) case.

3. Area- and Volume-Preserving Transformations

Area-preserving maps in finite-dimensional real Hilbert spaces $E$ (with parallelogram area $(a,b)$) are described by:

• In $\mathbb{R}^2$: $\varphi(a) = \varepsilon(a) A a$, with $|\det A| = 1$, $\varepsilon(a) \in \{-1,1\}$.
• In higher dimensions: only orthogonal linear operators $R$ up to sign functions, $\varphi(a) = \varepsilon(a) R a$ (Gehér, 2014).

This sits within the tradition of Wigner's theorem (inner product preservers), Mazur-Ulam (isometries), and deepens connections to operator theory: maps that preserve commutator norms between self-adjoint operators are forced to have pre-determined algebraic form.

4. Group-Theoretic and Algebraic Form-Preserving Transformations

Transformations preserving geometric or combinatorial form (e.g., adjacency/closeness) on products of Grassmann spaces or Grassmannians must be induced by semilinear automorphisms or dualities—almost always exclusively—unless in "trivial" or exceptional dimensions (Havlicek et al., 2013, Pankov, 2020).

On products $\mathcal{G}$ of complementary subspaces, all adjacency-preserving bijections are restrictions of automorphisms of the underlying vector space.

In Hilbert Grassmannians (subspace sets), injective orthogonality-preserving maps are always induced by (anti-)unitary operators, save for finitely many exceptional cases in high dimension; in infinite dimension, this is true only under extra spanning/maximality conditions (Pankov, 2020).

5. Form-Preserving Transformations for Partial Differential Equations

Families of differential equations often admit form-preserving equivalence transformations, which correspond to symmetries and allow classification and canonical reduction (Ndogmo, 2011, Cherniha et al., 2013):

• For families defined by arbitrary functions $\mathcal{A}(x)$, the equivalence group is always a subgroup of that for families with extended dependencies $\mathcal{A}(x, y^{(s)})$.
• For reaction-diffusion systems with constant diffusivities, non-degenerate form-preserving transformations are parameterized by scaling, translation, and exponential weights on the dependent variables, subject to functional constraints ensuring mapping within the class.
• These transformations are central in constructing exact solutions via $Q$-conditional symmetries; the framework is strict enough to guarantee that classification tables of PDEs (e.g., two-component systems admit exactly 26 non-equivalent forms) cannot be reduced further up to these transformations (Cherniha et al., 2013).

6. Moment-Sequence and Positive Matrix Preservers

Functions preserving the set of moment sequences under entrywise action are classified:

• On $[0,1]$ or $[−1,1]$ (moment sequences of measures), only absolutely monotonic (all derivatives nonnegative) analytic (entire) functions are permitted (Belton et al., 2016).
• For Hankel matrices of bounded rank, preservation on rank ≤3 implies global preservation.
• In the totally non-negative matrix case, only constant or linear maps qualify—demonstrating extreme rigidity.

Multivariable generalization introduces "facewise absolutely monotonic" functions: on each orthant face where some variables vanish, the restriction must be compatible and absolutely monotonic in the remaining variables.

7. Form-Preserving Transformations in Quantum and Phase-Space Mechanics

In TDSE (Schrödinger equations), explicit form-preserving transformations—space/time rescalings, translations, phases—map solutions between different potentials, often linking free and force-affected dynamics or relating coherent/excited state evolutions: $$x' = \frac{x}{\gamma(t)} + \beta(t),\quad t' = \int^t \frac{ds}{\gamma^2(s)},\quad \psi'(x', t') = \sqrt{\gamma(t)}\,\psi(x,t) \, e^{\text{quadratic phase}}$$ with induced potential changes calculated algebraically (Amin et al., 24 Oct 2025).

Phase-space (Wigner function) dynamics show that such transformations induce canonical (affine) transformations—Wigner functions behave as classical probability densities under these changes, and the associated quantum (Moyal) bracket structure is preserved.

8. Structure-Preserving Transformations in Matrix Analysis

Normal structured matrices (Hamiltonian, per-Hermitian, etc.) admit explicit canonical forms under unitary structure-preserving transformations:

• Hamiltonian: block-diagonal forms via unitary symplectic similarity
• Per-Hermitian: symmetric, per-symmetric real blocks
• Skew-this or per-skew-that: reductions via multiplication by $i$

Such transformations are absolutely essential for numerically stable, physically valid canonical forms and for algorithms in spectral theory and control systems (Begovic et al., 2018).

9. Form-Preserving Transformations in Geometric Invariant Theory

In affine geometry, the invariants under form-preserving transformations—actions of $SL(d), GL(d), Aff(d)$—are ratios of top forms (determinants/wedge products) of position or relative (Jacobi) coordinates. These quantities generalize the area, volume, and higher-dimensional geometric invariants, and are constructed as explicit solutions to invariance PDEs (Anderson, 2018): $$f\left( \frac{ \bigwedge_{a=1}^d {\bm \rho}^{i_a} }{ \bigwedge_{a=1}^d {\bm \rho}^{j_a} } \right)$$ for relational volume elements.

10. Synthesis and Further Directions

Across all domains, form-preserving transformations exhibit strong rigidity, being classified almost always by analytic or algebraic structure (linear, affine, projective, automorphic, or group-theoretic), supplemented in analytic contexts by monotonicity or symplectic constraints.

This universal structure manifests in quantum measurement theory, functional analysis, operator algebras, geometry, matrix theory, PDE symmetry analysis, and quantum dynamics. Any allowed freedom corresponds either to intrinsic symmetries (automorphism groups, isometries, projectivities) or, in analytic cases, to monotonicity/analyticity constraints.

Form-preserving transformations thus underpin classification theorems, canonical forms, invariance theory, and constructive solution techniques in diverse areas of mathematics, mathematical physics, and applications.