Spinorial Representation of Elastic Surface and Membrane Vibrations: From Extrinsic Curvature Model to Spin-Connected Dirac-like Operator.
POSTER
Abstract
We present a unified geometric framework that connects classical elastic-surface vibrations to spinor dynamics on curved manifolds. Starting from the intrinsic wave equation for small deformations of a spherical surface,
∂t2h+μh−TΔsh+κΔs2h=0
we identify the fourth-order bending operator Δs2 as the square of a Dirac-type operator D2 = -Δs + R/4 on the sphere S2(R). This correspondence reveals that the square root of the bending term generates a Schrödinger-like evolution, while the square root of the stretching term yields a fractional Laplacian dispersion. Analytical spectra for both operators are derived, showing curvature-dependent shifts ωl= (l + 1)/R and degeneracies linked to spin connection geometry. By coupling the scalar displacement field h(θ,φ,t) to a two-component spinor field Ψ(θ,φ,t) through intrinsic covariant derivatives, we construct a consistent field-theoretic model unifying curvature, orientation, and oscillation. This spinorial representation replaces extrinsic bending constraints with intrinsic orientation encoded by the spin connection, offering new insights into curvature-mediated excitations relevant to strained graphene, topological mechanics, and soft-matter membranes.
∂t2h+μh−TΔsh+κΔs2h=0
we identify the fourth-order bending operator Δs2 as the square of a Dirac-type operator D2 = -Δs + R/4 on the sphere S2(R). This correspondence reveals that the square root of the bending term generates a Schrödinger-like evolution, while the square root of the stretching term yields a fractional Laplacian dispersion. Analytical spectra for both operators are derived, showing curvature-dependent shifts ωl= (l + 1)/R and degeneracies linked to spin connection geometry. By coupling the scalar displacement field h(θ,φ,t) to a two-component spinor field Ψ(θ,φ,t) through intrinsic covariant derivatives, we construct a consistent field-theoretic model unifying curvature, orientation, and oscillation. This spinorial representation replaces extrinsic bending constraints with intrinsic orientation encoded by the spin connection, offering new insights into curvature-mediated excitations relevant to strained graphene, topological mechanics, and soft-matter membranes.
Presenters
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Amar Dadel
- Miami University