An additive calculus for almost submetries

Definitions and mathematical statement

A submetry is a map $f:X\to Y$ taking every open metric ball $B_X(x,r)$ onto $B_Y(f(x),r)$. The source asks to study Alexandrov (almost) submetries without choosing a definition of almost or a theorem. The supplied additive model is as follows. Let $f$ be surjective and let $\alpha,\beta\geq0$. Require, for all $x,x'\in X$ and $y\in Y$, \[ d_Y(f(x),f(x'))\leq d_X(x,x')+\alpha,\qquad \operatorname{dist}_X(x,f^{-1}(y))\leq d_Y(f(x),y)+\beta. \] The first condition is additive shortness; the second is additive ball lifting. Using infimal distance avoids assuming closest points exist. Let $\Delta(f)=\sup_y\operatorname{diam}_X f^{-1}(y)<\infty$. When $\alpha=\beta=0$ these conditions give the open-ball submetry property. For positive additive error the map is not necessarily Lipschitz or continuous.

The supplied quantitative theorem, valid for arbitrary metric spaces, is \[ d_Y(fx,fx')-\alpha\leq d_X(x,x')\leq d_Y(fx,fx')+\beta+\Delta(f). \] Consequently the graph correspondence $R_f=\{(x,f(x)):x\in X\}$ has \[ \operatorname{dis}(R_f)=\sup_{x,x'}|d_X(x,x')-d_Y(fx,fx')| \leq\max\{\alpha,\beta+\Delta(f)\}. \] For compact $X,Y$, surjectivity makes this a correspondence and gives $d_{\mathrm{GH}}(X,Y)\leq\tfrac12\max\{\alpha,\beta+\Delta(f)\}$. A large-fiber exact submetry is not asserted to be a small GH approximation.

For a finite tower of surjections $f_i:X_{i-1}\to X_i$, $1\leq i\leq m$, with errors $\alpha_i,\beta_i$ and fiber bounds $\Delta_i$, let $F=f_m\circ\cdots\circ f_1$. The aid gives additive shortness and lifting errors $\sum_i\alpha_i$ and $\sum_i\beta_i$, the telescoping distortion bound \[ \operatorname{dis}(R_F)\leq \max\left\{\sum_{i=1}^m\alpha_i,\sum_{i=1}^m(\beta_i+\Delta_i)\right\}, \] and the separate terminal-fiber estimate \[ \Delta(F)\leq\sum_{i=1}^m\Delta_i+\sum_{i=1}^{m-1}\beta_i. \] The final-stage lifting error is absent from this fiber bound. For compact endpoints the GH bound is half the displayed distortion estimate. These metric lemmas apply in particular when the spaces are Alexandrov, but do not by themselves establish curvature preservation, singular-stratum structure, or a geometric classification of almost submetries. The source gives no reason to identify this additive convention with every usage of that term.

Short English statement

The supplied interpretation measures additive shortness and ball-lifting defects separately. Together with fiber diameters they control graph distortion and compact Gromov--Hausdorff distance, with explicit finite-composition bounds. The results are general metric estimates applicable to Alexandrov spaces, not a full Alexandrov classification or a unique definition of almost submetry.

Sources

• Local registry: permanent catalogue entry, exact source code catalogue entry; aim-topology-notes.json, notes[228].
• AIM workshop Manifolds with nonnegative sectional curvature, Problem 16, https://aimath.org/WWN/nnsectcurvature/nnsectcurvature.pdf. No proposer, year, remarks, or original literature reference is supplied.
• Verification reports the same sentence in the official PDF and no apparent OCR error, while explicitly noting the absence of a definition of almost submetry or a specified theorem.
• Supplied research attempt 1, registry updated 14 August 2026: partial result, low novelty confidence, machine-generated and not peer reviewed. It supplies the two-defect additive definition and telescoping estimates. No complete authoritative external citation is included.