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 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].