A generalization of manifolds with corners

Abstract

In conventional Differential Geometry one studies manifolds, locally modelled on Rn, manifolds with boundary, locally modelled on [0,∞)×Rn−1, and manifolds with corners, locally modelled on [0,∞)k×Rn−k. They form categories Man⊂Manb⊂Manc. Manifolds with corners X have boundaries ∂X, also manifolds with corners, with dim∂X=dimX−1.We introduce a new notion of manifolds with generalized corners, or manifolds with g-corners, extending manifolds with corners, which form a category Mangc with Man⊂Manb⊂Manc⊂Mangc. Manifolds with g-corners are locally modelled on XP=HomMon(P,[0,∞)) for P a weakly toric monoid, where XP≅[0,∞)k×Rn−k for P=Nk×Zn−k.Most differential geometry of manifolds with corners extends nicely to manifolds with g-corners, including well-behaved boundaries ∂X. In some ways manifolds with g-corners have better properties than manifolds with corners; in particular, transverse fibre products in Mangc exist under much weaker conditions than in Manc.This paper was motivated by future applications in symplectic geometry, in which some moduli spaces of J-holomorphic curves can be manifolds or Kuranishi spaces with g-corners rather than ordinary corners.Our manifolds with g-corners are related to the ‘interior binomial varieties’ of Kottke and Melrose [20], and the ‘positive log differentiable spaces’ of Gillam and Molcho [6].