# Manifolds

While the interface `ManifoldsBase.jl`

does not cover concrete manifolds, it provides a few helpers to build or create manifolds based on existing manifolds

## A default manifold

`DefaultManifold`

is a simplified version of `Euclidean`

and demonstrates a basic interface implementation. It can be used to perform simple tests. Since when using `Manifolds.jl`

the `Euclidean`

is available, the `DefaultManifold`

itself is not exported.

`ManifoldsBase.DefaultManifold`

— Type`DefaultManifold <: AbstractManifold`

This default manifold illustrates the main features of the interface and provides a skeleton to build one's own manifold. It is a simplified/shortened variant of `Euclidean`

from `Manifolds.jl`

.

This manifold further illustrates how to type your manifold points and tangent vectors. Note that the interface does not require this, but it might be handy in debugging and educative situations to verify correctness of involved variables.

**Constructor**

`DefaultManifold(n::Int...; field = ℝ, parameter::Symbol = :field)`

Arguments:

`n`

: shape of array representing points on the manifold.`field`

: field over which the manifold is defined. Either`ℝ`

,`ℂ`

or`ℍ`

.`parameter`

: whether a type parameter should be used to store`n`

. By default size is stored in a field. Value can either be`:field`

or`:type`

.

## Embedded manifold

The embedded manifold is a manifold $\mathcal M$ which is modelled *explicitly* specifying its embedding $\mathcal N$ in which the points and tangent vectors are represented. Most prominently `is_point`

and `is_vector`

of an embedded manifold are implemented to check whether the point is a valid point in the embedding. This can of course still be extended by further tests. `ManifoldsBase.jl`

provides two possibilities of easily introducing this in order to dispatch some functions to the embedding.

### Implicit case: the `IsEmbeddedManifold`

Trait

For the implicit case, your manifold has to be a subtype of the `AbstractDecoratorManifold`

. Adding a method to the `active_traits`

function for a manifold that returns an `AbstractTrait`

`IsEmbeddedManifold`

, makes that manifold an embedded manifold. You just have to also define `get_embedding`

so that appropriate functions are passed on to that embedding. This is the implicit case, since the manifold type itself does not carry any information about the embedding, just the trait and the function definition do.

### Explicit case: the `EmbeddedManifold`

The `EmbeddedManifold`

itself is an `AbstractDecoratorManifold`

so it is a case of the implicit embedding itself, but internally stores both the original manifold and the embedding. They are also parameters of the type. This way, an additional embedding of one manifold in another can be modelled. That is, if the manifold is implemented using the implicit embedding approach from before but can also be implemented using a *different* embedding, then this method should be chosen, since you can dispatch functions that you want to implement in this embedding then on the type which explicitly has the manifold and its embedding as parameters.

Hence this case should be used for any further embedding after the first or if the default implementation works without an embedding and the alternative needs one.

`ManifoldsBase.EmbeddedManifold`

— Type`EmbeddedManifold{𝔽, MT <: AbstractManifold, NT <: AbstractManifold} <: AbstractDecoratorManifold{𝔽}`

A type to represent an explicit embedding of a `AbstractManifold`

`M`

of type `MT`

embedded into a manifold `N`

of type `NT`

. By default, an embedded manifold is set to be embedded, but neither isometrically embedded nor a submanifold.

This type is not required if a manifold `M`

is to be embedded in one specific manifold `N`

. One can then just implement `embed!`

and `project!`

. You can further pass functions to the embedding, for example, when it is an isometric embedding, by using an `AbstractDecoratorManifold`

. Only for a second –maybe considered non-default– embedding, this type should be considered in order to dispatch on different embed and project methods for different embeddings `N`

.

**Fields**

`manifold`

the manifold that is an embedded manifold`embedding`

a second manifold, the first one is embedded into

**Constructor**

`EmbeddedManifold(M, N)`

Generate the `EmbeddedManifold`

of the `AbstractManifold`

`M`

into the `AbstractManifold`

`N`

.

`ManifoldsBase.decorated_manifold`

— Method`decorated_manifold(M::EmbeddedManifold, d::Val{N} = Val(-1))`

Return the manifold of `M`

that is decorated with its embedding. For this specific type the internally stored enhanced manifold `M.manifold`

is returned.

See also `base_manifold`

, where this is used to (potentially) completely undecorate the manifold.

`ManifoldsBase.get_embedding`

— Method`get_embedding(M::EmbeddedManifold)`

Return the embedding `AbstractManifold`

`N`

of `M`

, if it exists.

## Metrics

Most metric-related functionality is currently defined in `Manifolds.jl`

but a few basic types are defined here.

`ManifoldsBase.AbstractMetric`

— Type`AbstractMetric`

Abstract type for the pseudo-Riemannian metric tensor $g$, a family of smoothly varying inner products on the tangent space. See `inner`

.

**Functor**

```
(metric::Metric)(M::AbstractManifold)
(metric::Metric)(M::MetricManifold)
```

Generate the `MetricManifold`

that wraps the manifold `M`

with given `metric`

. This works for both a variable containing the metric as well as a subtype `T<:AbstractMetric`

, where a zero parameter constructor `T()`

is availabe. If `M`

is already a metric manifold, the inner manifold with the new `metric`

is returned.

`ManifoldsBase.EuclideanMetric`

— Type`EuclideanMetric <: RiemannianMetric`

A general type for any manifold that employs the Euclidean Metric, for example the `Euclidean`

manifold itself, or the `Sphere`

, where every tangent space (as a plane in the embedding) uses this metric (in the embedding).

Since the metric is independent of the field type, this metric is also used for the Hermitian metrics, i.e. metrics that are analogous to the `EuclideanMetric`

but where the field type of the manifold is `ℂ`

.

This metric is the default metric for example for the `Euclidean`

manifold.

`ManifoldsBase.RiemannianMetric`

— Type`RiemannianMetric <: AbstractMetric`

Abstract type for Riemannian metrics, a family of positive definite inner products. The positive definite property means that for $X ∈ T_p \mathcal M$, the inner product $g(X, X) > 0$ whenever $X$ is not the zero vector.

`ManifoldsBase.change_metric!`

— Method`change_metric!(M::AbstractcManifold, Y, G2::AbstractMetric, p, X)`

Compute the `change_metric`

in place of `Y`

.

`ManifoldsBase.change_metric`

— Method`change_metric(M::AbstractcManifold, G2::AbstractMetric, p, X)`

On the `AbstractManifold`

`M`

with implicitly given metric $g_1$ and a second `AbstractMetric`

$g_2$ this function performs a change of metric in the sense that it returns the tangent vector $Z=BX$ such that the linear map $B$ fulfills

\[g_2(Y_1,Y_2) = g_1(BY_1,BY_2) \quad \text{for all } Y_1, Y_2 ∈ T_p\mathcal M.\]

`ManifoldsBase.change_representer!`

— Method`change_representer!(M::AbstractcManifold, Y, G2::AbstractMetric, p, X)`

Compute the `change_metric`

in place of `Y`

.

`ManifoldsBase.change_representer`

— Method`change_representer(M::AbstractManifold, G2::AbstractMetric, p, X)`

Convert the representer `X`

of a linear function (in other words a cotangent vector at `p`

) in the tangent space at `p`

on the `AbstractManifold`

`M`

given with respect to the `AbstractMetric`

`G2`

into the representer with respect to the (implicit) metric of `M`

.

In order to convert `X`

into the representer with respect to the (implicitly given) metric $g_1$ of `M`

, we have to find the conversion function $c: T_p\mathcal M \to T_p\mathcal M$ such that

\[ g_2(X,Y) = g_1(c(X),Y)\]

## A manifold for validation

`ValidationManifold`

is a simple decorator using the `AbstractDecoratorManifold`

that “decorates” a manifold with tests that all involved points and vectors are valid for the wrapped manifold. For example involved input and output paratemers are checked before and after running a function, repectively. This is done by calling `is_point`

or `is_vector`

whenever applicable.

`ManifoldsBase.ValidationCoTVector`

— Type`ValidationCoTVector = ValidationFibreVector{CotangentSpaceType}`

Represent a cotangent vector to a point on an `ValidationManifold`

, i.e. on a manifold where data can be represented by arrays. The array is stored internally and semantically. This distinguished the value from `ValidationMPoint`

s vectors of other types.

`ManifoldsBase.ValidationFibreVector`

— Type`ValidationFibreVector{TType<:VectorSpaceType} <: AbstractFibreVector{TType}`

Represent a tangent vector to a point on an `ValidationManifold`

, i.e. on a manifold where data can be represented by arrays. The array is stored internally and semantically. This distinguished the value from `ValidationMPoint`

s vectors of other types.

`ManifoldsBase.ValidationMPoint`

— Type`ValidationMPoint <: AbstractManifoldPoint`

Represent a point on an `ValidationManifold`

, i.e. on a manifold where data can be represented by arrays. The array is stored internally and semantically. This distinguished the value from `ValidationTVector`

s and `ValidationCoTVector`

s.

`ManifoldsBase.ValidationManifold`

— Type`ValidationManifold{𝔽,M<:AbstractManifold{𝔽}} <: AbstractDecoratorManifold{𝔽}`

A manifold to encapsulate manifolds working on array representations of `AbstractManifoldPoint`

s and `TVector`

s in a transparent way, such that for these manifolds it's not necessary to introduce explicit types for the points and tangent vectors, but they are encapsulated/stripped automatically when needed.

This manifold is a decorator for a manifold, i.e. it decorates a `AbstractManifold`

`M`

with types points, vectors, and covectors.

**Constructor**

`ValidationManifold(M::AbstractManifold; error::Symbol = :error)`

Generate the Validation manifold, where `error`

is used as the symbol passed to all checks. This `:error`

s by default but could also be set to `:warn`

for example

`ManifoldsBase.ValidationTVector`

— Type`ValidationTVector = ValidationFibreVector{TangentSpaceType}`

Represent a tangent vector to a point on an `ValidationManifold`

, i.e. on a manifold where data can be represented by arrays. The array is stored internally and semantically. This distinguished the value from `ValidationMPoint`

s vectors of other types.

`ManifoldsBase.array_value`

— Method`array_value(p)`

Return the internal array value of an `ValidationMPoint`

, `ValidationTVector`

, or `ValidationCoTVector`

if the value `p`

is encapsulated as such. Return `p`

if it is already an array.