Orientability of metric manifolds and metric spaces with curvature bounds

Abstract

We develop a theory of orientability for metric manifolds and singular metric spaces using locally integral currents. For topological $n$-manifolds with locally finite Hausdorff measure, we introduce local degree and measure-theoretic conditions under which topological orientations correspond canonically to boundaryless, locally integer rectifiable $n$-currents of multiplicity one on the rectifiable part. We extend these currents to a class of singular spaces whose manifold part is dense and of full measure, and establish constancy and top-dimensional homology results under local uniqueness of orientation. For orientable non-collapsed $\RCD(κ,n)$ spaces, we prove a current-theoretic Stokes theorem: the boundary of the orientation current is the induced orientation current of the geometric boundary, with multiplicity one. In particular, its mass measure is precisely the $(n-1)$-dimensional Hausdorff measure on that boundary. For purely $n$-dimensional, locally geodesically complete spaces with local upper curvature bounds, we show that local uniqueness of orientation excludes codimension-one branching and implies a local $1$-Poincaré inequality. The latter result applies beyond homology manifolds and extends the connection between quantitative topology and Poincaré inequalities to this class of singular spaces.