Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 070, 6 pages      arXiv:1907.05610      https://doi.org/10.3842/SIGMA.2019.070
Contribution to the Special Issue on Algebra, Topology, and Dynamics in Interaction in honor of Dmitry Fuchs

Holomorphic Distributions and Connectivity by Integral Curves of Distributions

Vladimir A. Zorich
Division of Mathematical Analysis, Faculty of Mechanics and Mathematics,Lomonosov Moscow State University, GSP-1, Leninskie Gory, Moscow, 119991, Russia

Received July 19, 2019, in final form September 14, 2019; Published online September 19, 2019

Abstract
It is known that the classical Frobenius theorem on conditions of integrability for distributions of planes can be extended to the case of complex holomorphic distributions. We show that an alternative criterion for integrability, namely, non-connectivity, discovered (or at least, marked and explicitly formulated) by Carathéodory in relation to classical thermodynamics, also admits a holomorphic formulation.

Key words: holomorphic distribution; integral curve; connectivity; thermodynamic states; adiabatic transitions; Carathéodory's theorem.

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