An information-theoretic analog of a result of Perelman

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)−dim-ensional ball. 1 Manifolds 1.1 Topologies, coordinates, and measures Let k be a positive integer. By definition, a given compact topological space M is a manifold of k dimensions.if every point p ∈ M has a neighborhood that is topologically equivalent to a Euclidean open sphere, of k dimensions, centered at p. Call such a neighborhood a cell centered at p, .or, less formally, a cell. To a given cell C centered at p then corresponds a topological transformation T, specific to C, that transforms a Euclidean k-sphere, and therefore also transfers the coordinate axes defined therein, to a topological space C ⊂ M. On the Euclidean space T −1 C there exists a σ-algebra F E of measurable sets– the smallest σ-algebra that contains every open set. Because T takes open sets to open sets T F E is also a σ-algebra on C. Then T is what is often called a measurable transformation. It transfers the Lebesgue measure λ(·) defined on T −1 C to a measure µ(·) = λ(T −1 ·) on C. As the image of a Euclidean sphere, a cell centered at p can be given a quasi-Cartesian coordinate frame, specific to that cell, with origin at p and identified by its (curviliear) coordinates, say x 1 , x 2 , · · · , x k. M is compact and is covered by finitely many cells. Such a covering is a proper covering. Let C 1 , C 2 , be distinct cells of a given proper covering, endowed respectively with coordinate frames {x r }, {y s }. Let p be a point in C 1 ∩ C 2. Because M is a manifold, the coordinates {x r } of p are already constrained to be continuous 1