# Holomorphic Distributions and Connectivity by Integral Curves of Distributions

## TL;DR

This paper explores the extension of classical integrability criteria for complex holomorphic distributions, including a holomorphic reformulation of Carathéodory's non-connectivity criterion, linking differential geometry and thermodynamics.

## Contribution

It introduces a holomorphic version of Carathéodory's non-connectivity criterion, expanding the understanding of integrability conditions for complex distributions.

## Key findings

- Holomorphic extension of Carathéodory's non-connectivity criterion.
- Connection between integrability and non-connectivity in complex distributions.
- Bridging differential geometry with thermodynamic concepts.

## 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 Carath\'eodory in relation to classical thermodynamics, also admits a holomorphic formulation.