On manifolds with corners

Abstract

Manifolds without boundary, and manifolds with boundary, are universally known in Differential Geometry, but manifolds with corners (locally modelled on [0,\infty)^k x R^{n-k}) have received comparatively little attention. The basic definitions in the subject are not agreed upon, there are several inequivalent definitions in use of manifolds with corners, of boundary, and of smooth map, depending on the applications in mind. We present a theory of manifolds with corners which includes a new notion of smooth map f : X --> Y. Compared to other definitions, our theory has the advantage of giving a category Man^c of manifolds with corners which is particularly well behaved as a category: it has products and direct products, boundaries behave in a functorial way, and there are simple conditions for the existence of fibre products X x_Z Y in Man^c. Our theory is tailored to future applications in Symplectic Geometry, and is part of a project to describe the geometric structure on moduli spaces of J-holomorphic curves in a new way. But we have written it as a separate paper as we believe it is of independent interest.

34 pages. (v2) final version, to appear in proceedings of "The Conference on Geometry", in honour of S.-T. Yau, Advanced Lectures in Mathematics Series, International Press

Cited by in corpus (40)

• Non-Compact Symplectic Toric Manifolds
• Intersection theory on moduli of disks, open KdV and Virasoro
• Operational General Relativity: Possibilistic, Probabilistic, and Quantum
• The combinatorial formula for open gravitational descendents
• Exotic symplectic manifolds from Lefschetz fibrations
• Reductive group actions
• Single-valued integration and double copy
• The degenerate special Lagrangian equation
• D-manifolds, d-orbifolds and derived differential geometry: a detailed summary
• Topological Considerations for Tuning and Fingering Stringed Instruments
• Log differentiable spaces and manifolds with corners
• A new definition of Kuranishi space
• Moduli of Open Stable Maps to a Homogeneous Space
• Differential forms, Fukaya $A_{\infty }$ algebras, and Gromov-Witten axioms
• Source Spaces and Perturbations for Cluster Complexes
• Non-negative curvature and torus actions
• Faithful Semitoric Systems
• Indirect Influences on Directed Manifolds
• Some new homology and cohomology theories of manifolds and orbifolds
• Classification and equivariant cohomology of circle actions on 3d manifolds
• Manifolds with analytic corners
• Fixed-point Localization for ${\mathbb{RP}}^{2m}\subset {\mathbb{CP}}^{2m}$
• Equivariant Open Gromov-Witten Theory of $\mathbb{R}{\mathbb{P}}^{2m}\hookrightarrow \mathbb{C}{\mathbb{P}}^{2m}$
• Derivatives of Turing machines in Linear Logic
• SG-Lagrangian submanifolds and their parametrization
• Configuration space integrals and the topology of knot and link spaces
• Building the bicategory Span${}_2(\mathcal{C})$
• Sunspot Equilibrium in General Quitting Games
• On the rapid decay homology of F.Pham
• Differential forms on orbifolds with corners
• IBL-Infinity Model of String Topology from Perturbative Chern-Simons Theory
• Tangential thickness of manifolds
• Symplectic toric stratified spaces with isolated singularities
• Analytic torsion on manifolds with fibred boundary metrics
• Toric contact geometry in arbitrary codimension
• Gluing Manifolds in the Cahiers Topos
• Poisson structure on manifolds with corners
• Vertical Genera
• Morse Homotopy and Homological Conformal Field Theory
• Transversality theorem in highly relative situations and its application