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Register a new userIndicatrix geometry clarifies that Finsler length can be larger than relative length
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Authors
E. Minguzzi
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I show that Matsumoto conjectured inequality between relative length and Finsler length is false. The incorrectness of the claim is easily inferred from the geometry of the indicatrix.
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In this work we consider an extraordinary quantum mechanical effect when, roughly speaking, the nucleus of an atom becomes (linearly) larger than the whole atom. Precisely, we consider Helium ion (in the ground state of the electron) moving translationally with the speed much smaller than speed of the electron rotation. This translation, effectively, changes neither the total momentum, nor the de Broglie wave length of the electron, nor the linear size of the atom corresponding to the diameter of the electron orbit. But, this translation implies a small nucleus momentum and nuclear de Broglie wavelength almost hundred times larger than the electron de Broglie wavelength. In the measurement of the nucleus wavelength using a diffraction apparatus with a characteristic length constant proportional to the proposed nucleus wavelength, according to standard quantum mechanical formalism, the nucleus behaves practically certainly as a wave. Then the unique, irreducible linear characteristic size for such a nucleus is de Broglie wavelength. Such a measurement effectively influences neither the electron dynamics nor linear size of the atom. This implies that, in such measurement, the size of the nucleus is in one dimension larger than the whole atom, i.e. electron orbital. All this corresponds metaphorically to the famous Leonardo fresco Last Supper where Jesus words coming from the nucleus, i.e. center of the composition, cause an expanding superposition or dramatic wave-like movement of the apostles.
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A geodesic circle in Finsler geometry is a natural extension of that in a Euclidean space. In this paper, we apply Lie derivatives and the Cartan $Y$-connection to study geodesic circles and (infinitesimal) concircular transformations on a Finsler manifold. We characterize a concircular vector field with some PDEs on the tangent bundle, and then we obtain respective necessary and sufficient conditions for a concircular vector field to be conformal and a conformal vector field to be concircular. We also show conditions for two conformally related Finsler metrics to be concircular, and obtain some invariant curvature properties under conformal and concircular transformations.
In the standard approach to Finsler geometry the metric is defined as a vertical Hessian and the Chern or Cartan connections appear as just two among many possible natural linear connections on the pullback tangent bundle. Here it is shown that the Hessian nature of the metric, the non-linear connection and the Chern or Cartan connections can be derived from a few compatibility axioms between metric and Finsler connection. This result provides a metric foundation to Finsler geometry and hence justifies the claim that ``Finsler geometry is Riemannian geometry without the quadratic restriction. The paper also contains a study of the compatibility condition to be placed between the metric and the non-linear connection.
We show that any initial closed curve suitably close to a circle flows under length-constrained curve diffusion to a round circle in infinite time with exponential convergence. We provide an estimate on the total length of time for which such curves are not strictly convex. We further show that there are no closed translating solutions to the flow and that the only closed rotators are circles.
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