An information-theoretic analog of a result of Perelman

TL;DR

This paper introduces an information-theoretic analogue of Perelman's geometric result, demonstrating a measurable transformation from a compact manifold to a lower-dimensional surface of a higher-dimensional ball.

Contribution

It presents a novel connection between information theory and differential geometry by establishing a measurable transformation related to Perelman's theorem.

Findings

Existence of a measurable transformation from a compact manifold to a surface of a higher-dimensional ball.

Bridges concepts between information theory and geometric topology.

Provides a new perspective on manifold embeddings using information-theoretic tools.

Abstract

Each compact manifold M of finite dimension k is differentiable and supports an intrinsic probability measure. There then exists a measurable transformation of M to the k-dimensional "surface" of the (k+1)-dimensional ball.