Constructing examples of manifolds over β #
We introduce the necessary bits to be able to define manifolds modelled over β^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 β (EuclideanSpace β (Fin n)) (EuclideanHalfSpace n)
for the model space used to definen
-dimensional real manifolds with boundaryModelWithCorners β (EuclideanSpace β (Fin n)) (EuclideanQuadrant n)
for the model space used to definen
-dimensional real manifolds with corners
Notations #
In the locale manifold
, we introduce the notations
π‘ n
for the identity model with corners onEuclideanSpace β (Fin n)
π‘β n
forModelWithCorners β (EuclideanSpace β (Fin n)) (EuclideanHalfSpace n)
.
For instance, if a manifold M
is boundaryless, smooth and modelled on EuclideanSpace β (Fin m)
,
and N
is smooth with boundary modelled on EuclideanHalfSpace n
, and f : M β N
is a smooth
map, then the derivative of f
can be written simply as mfderiv (π‘ m) (π‘β 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 quadrant in β^n
, used to model manifolds with corners, made of all vectors with nonnegative
coordinates.
Instances For
Definition of the model with corners (EuclideanSpace β (Fin n), EuclideanHalfSpace n)
, used as
a model for manifolds with boundary. In the locale manifold
, use the shortcut π‘β n
.
Instances For
Definition of the model with corners (EuclideanSpace β (Fin n), EuclideanQuadrant n)
, used as a
model for manifolds with corners
Instances For
The left chart for the topological space [x, y]
, defined on [x,y)
and sending x
to 0
in
EuclideanHalfSpace 1
.
Instances For
The right chart for the topological space [x, y]
, defined on (x,y]
and sending y
to 0
in
EuclideanHalfSpace 1
.
Instances For
Charted space structure on [x, y]
, using only two charts taking values in
EuclideanHalfSpace 1
.
The manifold structure on [x, y]
is smooth.
Register the manifold structure on Icc 0 1
, and also its zero and one.