Math 2.0

Global analysis, analysis on manifolds Definition

Manifold with corners

Builds on Topological spaceHausdorff spaceSecond-countable spaceHomeomorphismSmooth functionAtlasManifold with boundary

Definition

Let n≥1n \ge 1 be an integer and let ℝ‾+n={x∈ℝn:x1≥0,…,xn≥0}\overline{\R}^n_+ = \set{x \in \R^n : x_1 \ge 0, \ldots, x_n \ge 0} be the closed orthant, with the subspace topology. Let A⊆ℝnA \subseteq \R^n be any subset. A map f ⁣:A→ℝnf \colon A \to \R^n is smooth if each a∈Aa \in A has an open neighbourhood WW in ℝn\R^n with a smooth map g ⁣:W→ℝng \colon W \to \R^n equal to ff on A∩WA \cap W. Let MM be a topological space. A chart with corners on MM is a homeomorphism φ\varphi from an open subset U⊆MU \subseteq M onto an open subset of ℝ‾+n\overline{\R}^n_+. A smooth atlas with corners is a family of charts with corners whose domains cover MM. Its transition maps φj∘φi−1\varphi_j \circ \varphi_i^{-1} must be smooth in the above sense. A smooth manifold with corners of dimension nn is a space MM such that:

  1. MM is Hausdorff.
  2. MM is second-countable.
  3. MM carries a maximal smooth atlas with corners of dimension nn.

Remarks

The square [0,1]2[0, 1]^2 is a manifold with corners. Near the vertex (0,0)(0, 0) the identity is a chart onto an open subset of ℝ‾+2\overline{\R}^2_+. Near (1,1)(1, 1) the chart is (x,y)↦(1−x,1−y)(x, y) \mapsto (1 - x, 1 - y), and near the other points one uses similar affine maps. All transition maps are affine, hence smooth, and the charts generate a maximal atlas.

The solid square pyramid in ℝ3\R^3 is not a manifold with corners for this structure, because clause 3 fails at its apex. The tangent cone of the pyramid at the apex has four faces. The derivative of a chart at the apex would map it linearly onto the cone ℝ‾+3\overline{\R}^3_+, which has only three faces.

A chart with corners is in particular a chart in the sense of an atlas modelled on ℝ‾+n\overline{\R}^n_+. As a topological space, every manifold with corners is a manifold with boundary, because ℝ‾+n\overline{\R}^n_+ is homeomorphic to the half-space {xn≥0}\set{x_n \ge 0}. In Mathlib, manifolds with corners are modelled through a ModelWithCorners.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

Show the path as a graph
Atlas, concept, no evidence levelAtlasconceptHausdorff space, concept, no evidence levelHausdorff spaceconceptHomeomorphism, concept, no evidence levelHomeomorphismconceptManifold with boundary, concept, no evidence levelManifold with boundaryconceptSecond-countable space, concept, no evidence levelSecond-countable spaceconceptSmooth function, concept, no evidence levelSmooth functionconceptTopological space, concept, no evidence levelTopological spaceconceptManifold with corners, definition, no evidence levelManifold with cornersdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Atlas concept
  2. Hausdorff space concept
  3. Homeomorphism concept
  4. Manifold with boundary concept
  5. Second-countable space concept
  6. Smooth function concept
  7. Topological space concept
  8. Manifold with corners definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Manifold
About this page
TagEQDDVAKSVE
URIm2://EQDDVAKSVE
Kinddefinition
WritingW0 stub
MSC58A05, 57R99
DefinesManifold with corners
UsesTopological space, Hausdorff space, Second-countable space, Homeomorphism, Smooth function, Atlas, Manifold with boundary
Changed2 November 2026

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