The Curvature Fingerprint ✛ ✎

Original: "Maximal Normal Curvature and Veronese Rigidity"

arXiv:2607.00949 · 2026-07-01 · arXiv non-exclusive · independent analysis · 1 min

A single fundamental form sets a bound on surface bending, and when it reaches equality, it completely determines the surface.

A geometer presses an invisible plate against a curved surface, and by how it bends, takes its fingerprint. That fingerprint isn't just a pattern; it's a number: the maximum curvature of all curves passing through a point. But what if the fingerprint is left by the surface's own intrinsic structure: its 'fundamental form' (a hidden geometric object that governs bending)? New work shows: when this form is harmonic (evenly distributed, stress-free), it rigidly dictates how much the surface can bend.

The key to this discovery is the union of ideas from Carl Friedrich Gauss and Bernhard Riemann. Gauss derived differential equations linking extrinsic bending to intrinsic properties. Riemann advanced curvature analysis, and Spectral Analysis of the Bochner formula turned this into a sharp inequality. The Variational Method yields an integral identity: the sum of the form's energy and a curvature term equals zero. After an optimization estimate, a nonlinear dynamics bound emerges: the maximum curvature cannot be less than √(2n/(n+1)). For n=2, that's roughly 1,155—only 15% more than the curvature of a round sphere.

Thus, the harmonic 'fingerprint' doesn't just set a bound: it uniquely identifies the immersion. The curvature-minimizing surfaces are exactly the projective planes with their canonical geometry. A complete topology classification.

The Bochner–Gauss method turns intrinsic harmony into extrinsic rigidity. Traditionally, this technique was used to analyze global structure; now it serves as a bridge between forms and the geometry of submanifolds. Perhaps other 'fingerprints'—closed forms of special types — obey similarly strict universal bounds. This is a step toward a general Dynamical Systems of rigidity for immersions.

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The Veronese embedding, discovered back in the 19th century, today helps construct optimal spherical codes and plays a role in quantum computing.

\kappa(F) \ge \sqrt{\frac{2n}{n+1}}

the smallest possible maximum curvature for immersions with a harmonic structure

0 = \int_\Sigma |\nabla \Psi|^2 d\mu + \int_\Sigma \langle \sum_\alpha q^{(p)}_{A_\alpha} \Psi, \Psi \rangle d\mu

links the energy of the form and the extrinsic curvature

Scientists

George Gabriel Stokes Henri Poincaré

Concepts

Archimedes' principle uniform convergence supersymmetry surface roughness formula

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