# Documentation

Mathlib.Geometry.Manifold.Instances.Real

# Constructing examples of manifolds over ℝ #

We introduce the necessary bits to be able to define manifolds modelled over ℝ^n, boundaryless or with boundary or with corners. As a concrete example, we construct explicitly the manifold with boundary structure on the real interval [x, y].

More specifically, we introduce

• ModelWithCorners ℝ (EuclideanSpace ℝ (Fin n)) (EuclideanHalfSpace n) for the model space used to define n-dimensional real manifolds with boundary
• ModelWithCorners ℝ (EuclideanSpace ℝ (Fin n)) (EuclideanQuadrant n) for the model space used to define n-dimensional real manifolds with corners

## Notations #

In the locale manifold, we introduce the notations

• 𝓡 n for the identity model with corners on EuclideanSpace ℝ (Fin n)
• 𝓡∂ n for ModelWithCorners ℝ (EuclideanSpace ℝ (Fin n)) (EuclideanHalfSpace n).

For instance, if a manifold M is boundaryless, smooth and modelled on EuclideanSpace ℝ (Fin m), and N is smooth with boundary modelled on EuclideanHalfSpace n, and f : M → N is a smooth map, then the derivative of f can be written simply as mfderiv (𝓡 m) (𝓡∂ n) f (as to why the model with corners can not be implicit, see the discussion in smooth_manifold_with_corners.lean).

## Implementation notes #

The manifold structure on the interval [x, y] = Icc x y requires the assumption x < y as a typeclass. We provide it as [Fact (x < y)].

The half-space in ℝ^n, used to model manifolds with boundary. We only define it when 1 ≤ n, as the definition only makes sense in this case.

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The quadrant in ℝ^n, used to model manifolds with corners, made of all vectors with nonnegative coordinates.

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theorem EuclideanQuadrant.ext {n : } (x : ) (y : ) (h : x = y) :
x = y
theorem EuclideanHalfSpace.ext {n : } [Zero (Fin n)] (x : ) (y : ) (h : x = y) :
x = y
theorem range_half_space (n : ) [Zero (Fin n)] :
(Set.range fun x => x) = {y | 0 y 0}
theorem range_quadrant (n : ) :
(Set.range fun x => x) = {y | ∀ (i : Fin n), 0 y i}

Definition of the model with corners (EuclideanSpace ℝ (Fin n), EuclideanHalfSpace n), used as a model for manifolds with boundary. In the locale manifold, use the shortcut 𝓡∂ n.

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Definition of the model with corners (EuclideanSpace ℝ (Fin n), EuclideanQuadrant n), used as a model for manifolds with corners

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def IccLeftChart (x : ) (y : ) [h : Fact (x < y)] :
LocalHomeomorph (↑(Set.Icc x y)) ()

The left chart for the topological space [x, y], defined on [x,y) and sending x to 0 in EuclideanHalfSpace 1.

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def IccRightChart (x : ) (y : ) [h : Fact (x < y)] :
LocalHomeomorph (↑(Set.Icc x y)) ()

The right chart for the topological space [x, y], defined on (x,y] and sending y to 0 in EuclideanHalfSpace 1.

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instance IccManifold (x : ) (y : ) [h : Fact (x < y)] :
ChartedSpace () ↑(Set.Icc x y)

Charted space structure on [x, y], using only two charts taking values in EuclideanHalfSpace 1.

instance Icc_smooth_manifold (x : ) (y : ) [Fact (x < y)] :

The manifold structure on [x, y] is smooth.

Register the manifold structure on Icc 0 1, and also its zero and one.