P Symphonic Morphisms

• P symphonic morphism is a smooth map that minimizes a nonlinear p-symphonic energy integral, generalizing harmonic morphisms.
• The framework integrates variational methods and nonlinear PDE analysis to establish existence, rigidity, and classification results.
• Applications span geometric analysis and even musical transformation theory, linking combinatorial and algebraic structures with geometry.

A P Symphonic Morphism is a significant generalization of the symphonic morphism concept, formulated for the setting of nonlinear variational problems on maps between Riemannian manifolds, with crucial connections to geometric analysis and deep analogies with the theory of harmonic morphisms. The key definitions, existence results, and classification theorems for P symphonic morphisms reside at the intersection of nonlinear PDE theory, geometric mapping problems, and (in certain contexts) the algebraic representation of musical or combinatorial transformations.

1. Definition and Variational Framework

A P symphonic morphism (or $p$-symphonic morphism) is a smooth map $u : (M, g) \to (N, h)$ between Riemannian manifolds that critically minimizes the nonlinear $p$-symphonic energy

$E_p(u) = \int_M |u^* h|^p \, dv_g,$

with $|u^* h|$ defined as the Hilbert-Schmidt norm of the pullback metric. The Euler–Lagrange equation resulting from the first variation is

$$\operatorname{div}_g\left( (u^*h)^{p-2} \sigma_u \right) = 0,$$

where $\sigma_u(X) = \sum_i h(du(X), du(e_i))\, du(e_i)$, and $\{e_i\}$ is a local $g$-orthonormal frame (Cao, 14 Dec 2025). For $p=2$, the theory recovers that of classical symphonic morphisms (Cao, 2023); for general $p$, the framework extends to nonlinear analogues of harmonic morphisms.

A P symphonic morphism further satisfies a "function-preserving" property: for every local $p$-symphonic function $f: N \to \mathbb{R}$, the composition $f \circ u$ is itself $p$-symphonic, i.e.,

$$\operatorname{div}_g\left( |d(f\circ u)|^{p-2} \sigma_{f\circ u} \right) = 0,$$

whenever $f$ solves

$\operatorname{div}_h( |df|^{p-2} \sigma_f ) = 0.$

This property is precisely analogous to the defining property of harmonic morphisms, but for the nonlinear symphonic PDE (Cao, 14 Dec 2025).

2. Characterizations and Structural Results

P symphonic morphisms exhibit a structural parallel to the Fuglede–Ishihara characterization of harmonic morphisms. Sufficient conditions are encapsulated as follows:

• If $u$ is totally geodesic and horizontally weakly conformal with constant dilation $\lambda > 0$, then $u$ is a $p$-symphonic morphism for any $p \ge 2$.
• More precisely,

$$\operatorname{div}_g\left( (f\circ u)^*\tilde h^{\,p-2} \sigma_{f\circ u}\right) = \lambda^{2p} \left[\operatorname{div}_h\left( f^*\tilde h^{\,p-2} \sigma_f \right)\right] \circ u,$$

for every smooth $f: (N,h) \to (P,\tilde h)$ (Cao, 14 Dec 2025).

A partial converse also holds: if $u$ is a nonconstant $p$-symphonic morphism that is horizontally weakly conformal, then $u$ must be totally geodesic. The argument leverages the flexibility in prescribing jets of $p$-symphonic functions locally, which—via the morphism property—forces all components of the second fundamental form of $u$ to vanish.

These results generalize the link between horizontal conformality, vanishing second fundamental form, and the morphism property, now in the setting of nonlinear $p$-symphonic energies.

3. Generalizations and Connections

The P symphonic morphism framework supports several significant generalizations:

• Weighted metric–measure structures: The theory incorporates a weight function $\varphi : M \to \mathbb{R}$ and general $C^2$ functionals $F$, yielding the class of $\varphi$–$F$–symphonic maps with energy

$$E^{\varphi}_{sym}(u) = \int_M F\left(\frac{1}{2} |u^*h|^2 \right) e^{-\varphi} dv_g,$$

and Euler–Lagrange operator

$$\operatorname{div}_g( F'(\frac{1}{2}|u^*h|^2)\sigma(u) ) - \frac{1}{2} F'(\frac{1}{2}|u^*h|^2) |u^*h|^2 \nabla \varphi = 0$$

(Cao, 2023).

• Free boundary value problems: The analysis extends to domains with boundary, incorporating natural free-boundary conditions, with important rigidity theorems (Liouville type) for finite energy maps with limiting behavior at infinity (Cao, 2023).
• Higher-order and nonlinear energies: The theory interpolates between classical harmonic energies, $p$-harmonic map theories, and higher-order functionals (such as those studied by Han), thus unifying several strands of nonlinear geometric variational analysis (Cao, 14 Dec 2025, Cao, 2023).

4. Existence, Rigidity, and Examples

Existence Theorems: Existence of P symphonic morphisms is established for a broad class of geometric contexts. For instance, symphonic morphisms between ellipsoidal joins are constructed via either join or Hopf-type procedures. The central technical step is the reduction to a scalar fourth-order boundary-value ODE in the profile function $\varphi(t)$, handled via a coercive variational functional and Sobolev minimization, then regularity bootstrap. The existence of smooth solutions guarantees the construction in cases such as

$Q^{m_1+m_2+1}(a,b) \to Q^{n_1+n_2+1}(c,d)$

and analogous higher-dimensional generalizations (Cao, 2023).

Liouville Rigidity: For weighted $p$-symphonic energies with boundary, under weak growth conditions on the weight and conformal factor, every finite-energy map with prescribed boundary values at infinity must be constant. The proofs employ monotonicity formulas, weighted Bochner identities, and ODE comparison techniques to establish vanishing of energy in the large-scale, precluding nontrivial morphisms in such settings (Cao, 2023).

Examples:

Example	Description	Satisfies Morphism Conditions
Orthogonal projection	$$\pi: \mathbb{R}^m = \mathbb{R}^n \times \mathbb{R}^{m-n} \to \mathbb{R}^n$$ (standard, totally geodesic submersion)	$p$-symphonic for all $p \ge 2$
Hopf fibration	$S^{2k+1} \to \mathbb{CP}^k$ (Riemannian submersion with totally geodesic fibers)	$p$-symphonic for all $p$
Homothetic identity	$\mathrm{id}: (M, g) \to (M, c^2 g)$	$p$-symphonic for all $p$

(Cao, 14 Dec 2025)

Musical and Combinatorial Context: In musical transformation theory, the $P$ operator (parallel transformation in neo-Riemannian theory) acts as a “symphonic morphism” on pairs $(pc, \text{shape})$ (pitch class and major/minor), instantiated by a reflection in the dihedral group $D_{12} \cong \mathbb{Z}_{12} \rtimes \mathbb{Z}_2$. This large-scale mode switch preserves the pitch-class center while changing mode, providing an interpretive bridge between algebraic group actions and symphonic thematic development (Popoff, 2011).

5. Analogy with Harmonic Morphisms

P symphonic morphisms closely parallel the harmonic morphism theory, notably in:

• Functional preservation: Both pull back PDE solutions (harmonic or $p$-symphonic) from target to domain.
• Geometric prerequisites: Horizontal conformality and minimality/totally geodesic conditions are necessary and sufficient.
• Energy distinctions: The $p$-symphonic energies employ the pullback metric norm $|u^*h|^p$, diverging from the quadratic Dirichlet energy of harmonic maps.
• Nonlinear generality: For $p\ne2$, the resulting PDE is nonlinear, producing morphism classes distinct from harmonic morphisms except in special overlap (e.g., conformal submersions that are both harmonic and symphonic) (Cao, 14 Dec 2025).
• PDE theory: Local existence of $p$-symphonic functions is guaranteed under mild algebraic compatibility of prescribed derivatives (jets), via general elliptic theory (Cao, 14 Dec 2025).

6. Open Problems and Further Developments

Areas for further analysis and open problems include:

• Uniqueness and classification: Uniqueness of solutions to the nonlinear ODEs for profile functions in join/Hopf constructions is unresolved, even in rotationally symmetric settings (Cao, 2023).
• Extension to higher joins and warped targets: Scaling the constructions to joins of three or more spheres, or to more general warped-product targets beyond ellipsoids, remains open (Cao, 2023).
• Free-boundary and rigidity: The rigidity results derived from Liouville-type theorems indicate stringent constraints on nontrivial morphisms with free boundary and finite weighted energy; a complete classification for more general boundary settings is not yet available (Cao, 2023).
• Higher energy generalizations: The framework admits interpolation with $m$-harmonic or $S$-, $T$-energy functionals, potentially leading to a broader class of morphism theorems and associated geometric structures (Cao, 14 Dec 2025).
• Applications to special geometries: Synthesizing the symphonic morphism formalism with quasi-conformal, Kähler, or sub-Riemannian geometries for further curvature identities and mapping problems is a plausible direction.

7. Summary and Significance

The theory of P symphonic morphisms provides a nonlinear analogue and extension of harmonic morphism theory, grounded in the calculus of variations, elliptic PDE, and geometric mapping structures. It offers a rigorous framework for constructing and characterizing maps that preserve solutions to nonlinear variational PDEs under composition, with potent applications in geometry, analysis, and even algebraic structures such as group actions in music theory. Foundational existence and rigidity results are established under broad conditions, but comprehensive classification, uniqueness, and extension to more intricate geometric configurations remain active research areas (Cao, 14 Dec 2025, Cao, 2023, Cao, 2023, Popoff, 2011).