`ManifoldsBase.jl`

– an interface for manifolds

The interface for a manifold is provided in the lightweight package ManifoldsBase.jl. You can easily implement your algorithms and even your own manifolds just using the interface. All manifolds from the package here are also based on this interface, so any project based on the interface can benefit from all manifolds, as soon as a certain manifold provides implementations of the functions a project requires.

Additionally the `AbstractDecoratorManifold`

is provided as well as the `ValidationManifold`

as a specific example of such a decorator.

## Types and functions

The following functions are currently available from the interface. If a manifold that you implement for your own package fits this interface, we happily look forward to a Pull Request to add it here.

`ManifoldsBase.AbstractEstimationMethod`

— Type`AbstractEstimationMethod`

Abstract type for defining statistical estimation methods.

`ManifoldsBase.AbstractInverseRetractionMethod`

— Type`AbstractInverseRetractionMethod`

Abstract type for methods for inverting a retraction (see `inverse_retract`

).

`ManifoldsBase.AbstractRetractionMethod`

— Type`AbstractRetractionMethod`

Abstract type for methods for `retract`

ing a tangent vector to a manifold.

`ManifoldsBase.CoTVector`

— Type`CoTVector`

Type for a cotangent vector of a manifold. While a `Manifold`

does not necessarily require this type, for example when it is implemented for `Vector`

s or `Matrix`

type elements, this type can be used for more complicated representations, semantic verification, or even dispatch for different representations of cotangent vectors and their types on a manifold.

`ManifoldsBase.ExponentialRetraction`

— Type`ExponentialRetraction`

Retraction using the exponential map.

`ManifoldsBase.LogarithmicInverseRetraction`

— Type`LogarithmicInverseRetraction`

Inverse retraction using the `log`

arithmic map.

`ManifoldsBase.MPoint`

— Type`MPoint`

Type for a point on a manifold. While a `Manifold`

does not necessarily require this type, for example when it is implemented for `Vector`

s or `Matrix`

type elements, this type can be used for more complicated representations, semantic verification, or even dispatch for different representations of points on a manifold.

`ManifoldsBase.Manifold`

— Type`Manifold{F}`

A manifold type. The `Manifold`

is used to dispatch to different functions on a manifold, usually as the first argument of the function. Examples are the `exp`

onential and `log`

arithmic maps as well as more general functions that are built on them like the `geodesic`

.

The manifold is parametrized by an `AbstractNumbers`

to distinguish for example real (ℝ) and complex (ℂ) manifolds.

For subtypes the preferred order of parameters is: size and simple value parameters, followed by the `AbstractNumbers`

`field`

, followed by data type parameters, which might depend on the abstract number field type.

`ManifoldsBase.OutOfInjectivityRadiusError`

— Type`OutOfInjectivityRadiusError`

An error thrown when a function (for example `log`

arithmic map or `inverse_retract`

) is given arguments outside of its `injectivity_radius`

.

`ManifoldsBase.PolarInverseRetraction`

— Type`PolarInverseRetraction <: AbstractInverseRetractionMethod`

Inverse retractions that are based on a singular value decomposition of the matrix / matrices for point and tangent vector on a `Manifold`

`ManifoldsBase.PolarRetraction`

— Type`PolarRetraction <: AbstractRetractionMethod`

Retractions that are based on singular value decompositions of the matrix / matrices for point and tangent vector on a `Manifold`

`ManifoldsBase.ProjectionInverseRetraction`

— Type`ProjectionInverseRetraction <: AbstractInverseRetractionMethod`

Inverse retractions that are based on a projection (or its inversion).

`ManifoldsBase.ProjectionRetraction`

— Type`ProjectionRetraction <: AbstractRetractionMethod`

Retractions that are based on projection and usually addition in the embedding.

`ManifoldsBase.QRInverseRetraction`

— Type`QRInverseRetraction <: AbstractInverseRetractionMethod`

Inverse retractions that are based on a QR decomposition of the matrix / matrices for point and tangent vector on a `Manifold`

`ManifoldsBase.QRRetraction`

— Type`QRRetraction <: AbstractRetractionMethod`

Retractions that are based on a QR decomposition of the matrix / matrices for point and tangent vector on a `Manifold`

`ManifoldsBase.TVector`

— Type`TVector`

Type for a tangent vector of a manifold. While a `Manifold`

does not necessarily require this type, for example when it is implemented for `Vector`

s or `Matrix`

type elements, this type can be used for more complicated representations, semantic verification, or even dispatch for different representations of tangent vectors and their types on a manifold.

`Base.angle`

— Method`Base.exp`

— Method```
exp(M::Manifold, p, X)
exp(M::Manifold, p, X, t::Real = 1)
```

Compute the exponential map of tangent vector `X`

, optionally scaled by `t`

, at point `p`

from manifold the `Manifold`

`M`

.

`Base.isapprox`

— Method`isapprox(M::Manifold, p, X, Y; kwargs...)`

Check if vectors `X`

and `Y`

tangent at `p`

from `Manifold`

`M`

are approximately equal.

Keyword arguments can be used to specify tolerances.

`Base.isapprox`

— Method`isapprox(M::Manifold, p, q; kwargs...)`

Check if points `p`

and `q`

from `Manifold`

`M`

are approximately equal.

Keyword arguments can be used to specify tolerances.

`Base.log`

— Method`log(M::Manifold, p, q)`

Compute the logarithmic map of point `q`

at base point `p`

on the `Manifold`

`M`

.

`LinearAlgebra.norm`

— Method`ManifoldsBase.allocate`

— Method```
allocate(a)
allocate(a, dims::Integer...)
allocate(a, dims::Tuple)
allocate(a, T::Type)
allocate(a, T::Type, dims::Integer...)
allocate(a, T::Type, dims::Tuple)
```

Allocate an object similar to `a`

. It is similar to function `similar`

, although instead of working only on the outermost layer of a nested structure, it maps recursively through outer layers and calls `similar`

on the innermost array-like object only. Type `T`

is the new number element type `number_eltype`

, if it is not given the element type of `a`

is retained. The `dims`

argument can be given for non-nested allocation and is forwarded to the function `similar`

.

`ManifoldsBase.allocate_result`

— Method`allocate_result(M::Manifold, f, x...)`

Allocate an array for the result of function `f`

on `Manifold`

`M`

and arguments `x...`

for implementing the non-modifying operation using the modifying operation.

Usefulness of passing a function is demonstrated by methods that allocate results of musical isomorphisms.

`ManifoldsBase.allocate_result_type`

— Method`allocate_result_type(M::Manifold, f, args::NTuple{N,Any}) where N`

Return type of element of the array that will represent the result of function `f`

and the `Manifold`

`M`

on given arguments `args`

(passed as a tuple).

`ManifoldsBase.base_manifold`

— Function`base_manifold(M::Manifold, depth = Val(-1))`

Return the internally stored `Manifold`

for decorated manifold `M`

and the base manifold for vector bundles or power manifolds. The optional parameter `depth`

can be used to remove only the first `depth`

many decorators and return the `Manifold`

from that level, whether its decorated or not. Any negative value deactivates this depth limit.

`ManifoldsBase.check_manifold_point`

— Method`check_manifold_point(M::Manifold, p; kwargs...) -> Union{Nothing,String}`

Return `nothing`

when `p`

is a point on the `Manifold`

`M`

. Otherwise, return an error with description why the point does not belong to manifold `M`

.

By default, `check_manifold_point`

returns `nothing`

, i.e. if no checks are implemented, the assumption is to be optimistic for a point not deriving from the `MPoint`

type.

`ManifoldsBase.check_tangent_vector`

— Method`check_tangent_vector(M::Manifold, p, X; kwargs...) -> Union{Nothing,String}`

Check whether `X`

is a valid tangent vector in the tangent space of `p`

on the `Manifold`

`M`

. An implementation should first call `check_manifold_point(M, p; kwargs...)`

and then validate `X`

. If it is not a tangent vector, an error string should be returned.

By default, `check_tangent_vector`

returns `nothing`

, i.e. if no checks are implemented, the assumption is to be optimistic for tangent vectors not deriving from the `TVector`

type.

`ManifoldsBase.distance`

— Method`distance(M::Manifold, p, q)`

Shortest distance between the points `p`

and `q`

on the `Manifold`

`M`

.

`ManifoldsBase.embed!`

— Method`embed!(M::Manifold, Y, p, X)`

Embed a tangent vector `X`

at a point `p`

on the `Manifold`

`M`

into the ambient space and return the result in `Y`

. This method is only available for manifolds where implicitly an embedding or ambient space is given. Additionally, `embed!`

includes changing data representation, if applicable, i.e. if the tangents on `M`

are not represented in the same way as tangents on the embedding, the representation is changed accordingly. This is the case for example for Lie groups, when tangent vectors are represented in the Lie algebra. The embedded tangents are then in the tangent spaces of the embedded base points.

See also: `EmbeddedManifold`

, `project!`

`ManifoldsBase.embed!`

— Method`embed!(M::Manifold, q, p)`

Embed point `p`

from the `Manifold`

`M`

into the ambient space and return the result in `q`

. This method is only available for manifolds where implicitly an embedding or ambient space is given. Additionally, `embed`

includes changing data representation, if applicable, i.e. if the points on `M`

are not represented in the same way as points on the embedding, the representation is changed accordingly.

See also: `EmbeddedManifold`

, `project!`

`ManifoldsBase.embed`

— Method`embed(M::Manifold, p, X)`

Embed a tangent vector `X`

at a point `p`

on the `Manifold`

`M`

into the ambient space. This method is only available for manifolds where implicitly an embedding or ambient space is given. Additionally, `embed`

includes changing data representation, if applicable, i.e. if the tangents on `M`

are not represented in the same way as tangents on the embedding, the representation is changed accordingly. This is the case for example for Lie groups, when tangent vectors are represented in the Lie algebra. The embedded tangents are then in the tangent spaces of the embedded base points.

See also: `EmbeddedManifold`

, `project`

`ManifoldsBase.embed`

— Method`embed(M::Manifold, p)`

Embed point `p`

from the `Manifold`

`M`

into the ambient space. This method is only available for manifolds where implicitly an embedding or ambient space is given. Additionally, `embed`

includes changing data representation, if applicable, i.e. if the points on `M`

are not represented in the same way as points on the embedding, the representation is changed accordingly.

See also: `EmbeddedManifold`

, `project`

`ManifoldsBase.exp!`

— Method```
exp!(M::Manifold, q, p, X)
exp!(M::Manifold, q, p, X, t::Real = 1)
```

Compute the exponential map of tangent vector `X`

, optionally scaled by `t`

, at point `p`

from manifold the `Manifold`

`M`

. The result is saved to `q`

.

`ManifoldsBase.geodesic`

— Method`geodesic(M::Manifold, p, X) -> Function`

Get the geodesic with initial point `p`

and velocity `X`

on the `Manifold`

`M`

. The geodesic is the curve of constant velocity that is locally distance-minimizing. This function returns a function of (time) `t`

.

```
geodesic(M::Manifold, x, v, t::Real)
geodesic(M::Manifold, x, v, T::AbstractVector) -> AbstractVector
```

Return the point at time `t`

or points at times `t`

in `T`

along the geodesic.

`ManifoldsBase.injectivity_radius`

— Method`injectivity_radius(M::Manifold, p)`

Return the distance $d$ such that `exp(M, p, X)`

is injective for all tangent vectors shorter than $d$ (i.e. has an inverse).

`injectivity_radius(M::Manifold)`

Infimum of the injectivity radius of all manifold points.

```
injectivity_radius(M::Manifold[, x], method::AbstractRetractionMethod)
injectivity_radius(M::Manifold, x, method::AbstractRetractionMethod)
```

Distance $d$ such that `retract(M, p, X, method)`

is injective for all tangent vectors shorter than $d$ (i.e. has an inverse) for point `p`

if provided or all manifold points otherwise.

`ManifoldsBase.inner`

— Method`inner(M::Manifold, p, X, Y)`

Compute the inner product of tangent vectors `X`

and `Y`

at point `p`

from the `Manifold`

`M`

.

See also: `MetricManifold`

`ManifoldsBase.inverse_retract!`

— Method`inverse_retract!(M::Manifold, X, p, q[, method::AbstractInverseRetractionMethod])`

Compute the inverse retraction, a cheaper, approximate version of the `log`

arithmic map), of points `p`

and `q`

on the `Manifold`

`M`

. Result is saved to `X`

.

Inverse retraction method can be specified by the last argument, defaulting to `LogarithmicInverseRetraction`

. See the documentation of respective manifolds for available methods.

`ManifoldsBase.inverse_retract`

— Method```
inverse_retract(M::Manifold, x, y)
inverse_retract(M::Manifold, x, y, method::AbstractInverseRetractionMethod
```

Compute the inverse retraction, a cheaper, approximate version of the `log`

arithmic map), of points `p`

and `q`

on the `Manifold`

`M`

.

Inverse retraction method can be specified by the last argument, defaulting to `LogarithmicInverseRetraction`

. See the documentation of respective manifolds for available methods.

`ManifoldsBase.is_manifold_point`

— Function`is_manifold_point(M::Manifold, p, throw_error = false; kwargs...)`

Return whether `p`

is a valid point on the `Manifold`

`M`

.

If `throw_error`

is `false`

, the function returns either `true`

or `false`

. If `throw_error`

is `true`

, the function either returns `true`

or throws an error. By default the function calls `check_manifold_point(M, p; kwargs...)`

and checks whether the returned value is `nothing`

or an error.

`ManifoldsBase.is_tangent_vector`

— Function`is_tangent_vector(M::Manifold, p, X, throw_error = false; kwargs...)`

Return whether `X`

is a valid tangent vector at point `p`

on the `Manifold`

`M`

. Returns either `true`

or `false`

.

If `throw_error`

is `false`

, the function returns either `true`

or `false`

. If `throw_error`

is `true`

, the function either returns `true`

or throws an error. By default the function calls `check_tangent_vector(M, p, X; kwargs...)`

and checks whether the returned value is `nothing`

or an error.

`ManifoldsBase.log!`

— Method`log!(M::Manifold, X, p, q)`

Compute the logarithmic map of point `q`

at base point `p`

on the `Manifold`

`M`

. The result is saved to `X`

.

`ManifoldsBase.manifold_dimension`

— Method`manifold_dimension(M::Manifold)`

The dimension $n=\dim_{\mathcal M}$ of real space $\mathbb R^n$ to which the neighborhood of each point of the `Manifold`

`M`

is homeomorphic.

`ManifoldsBase.mid_point!`

— Method`ManifoldsBase.mid_point`

— Method`ManifoldsBase.number_eltype`

— Method`number_eltype(x)`

Numeric element type of the a nested representation of a point or a vector. To be used in conjuntion with `allocate`

or `allocate_result`

.

`ManifoldsBase.project!`

— Method`project!(M::Manifold, Y, p, X)`

Project ambient space representation of a vector `X`

to a tangent vector at point `p`

on the `Manifold`

`M`

. The result is saved in vector `Y`

. This method is only available for manifolds where implicitly an embedding or ambient space is given. Additionally, `project!`

includes changing data representation, if applicable, i.e. if the tangents on `M`

are not represented in the same way as points on the embedding, the representation is changed accordingly. This is the case for example for Lie groups, when tangent vectors are represented in the Lie algebra. after projection the change to the Lie algebra is perfomed, too.

See also: `EmbeddedManifold`

, `embed!`

`ManifoldsBase.project!`

— Method`project!(M::Manifold, q, p)`

Project point `p`

from the ambient space onto the `Manifold`

`M`

. The result is storedin `q`

. This method is only available for manifolds where implicitly an embedding or ambient space is given. Additionally, the projection includes changing data representation, if applicable, i.e. if the points on `M`

are not represented in the same array data, the data is changed accordingly.

See also: `EmbeddedManifold`

, `embed!`

`ManifoldsBase.project`

— Method`project(M::Manifold, p, X)`

Project ambient space representation of a vector `X`

to a tangent vector at point `p`

on the `Manifold`

`M`

. This method is only available for manifolds where implicitly an embedding or ambient space is given. Additionally, `project`

includes changing data representation, if applicable, i.e. if the tangents on `M`

are not represented in the same way as points on the embedding, the representation is changed accordingly. This is the case for example for Lie groups, when tangent vectors are represented in the Lie algebra. after projection the change to the Lie algebra is perfomed, too.

See also: `EmbeddedManifold`

, `embed`

`ManifoldsBase.project`

— Method`project(M::Manifold, p)`

Project point `p`

from the ambient space of the `Manifold`

`M`

to `M`

. This method is only available for manifolds where implicitly an embedding or ambient space is given. Additionally, the projection includes changing data representation, if applicable, i.e. if the points on `M`

are not represented in the same array data, the data is changed accordingly.

See also: `EmbeddedManifold`

, `embed`

`ManifoldsBase.representation_size`

— Method`representation_size(M::Manifold)`

The size of an array representing a point on `Manifold`

`M`

. Returns `nothing`

by default indicating that points are not represented using an `AbstractArray`

.

`ManifoldsBase.retract!`

— Method```
retract!(M::Manifold, q, p, X)
retract!(M::Manifold, q, p, X, t::Real=1)
retract!(M::Manifold, q, p, X, method::AbstractRetractionMethod)
retract!(M::Manifold, q, p, X, t::Real=1, method::AbstractRetractionMethod)
```

Compute a retraction, a cheaper, approximate version of the `exp`

onential map, from `p`

into direction `X`

, scaled by `t`

, on the `Manifold`

manifold `M`

. Result is saved to `q`

.

Retraction method can be specified by the last argument, defaulting to `ExponentialRetraction`

. See the documentation of respective manifolds for available methods.

`ManifoldsBase.retract`

— Method```
retract(M::Manifold, p, X)
retract(M::Manifold, p, X, t::Real=1)
retract(M::Manifold, p, X, method::AbstractRetractionMethod)
retract(M::Manifold, p, X, t::Real=1, method::AbstractRetractionMethod)
```

Compute a retraction, a cheaper, approximate version of the `exp`

onential map, from `p`

into direction `X`

, scaled by `t`

, on the `Manifold`

`M`

.

Retraction method can be specified by the last argument, defaulting to `ExponentialRetraction`

. See the documentation of respective manifolds for available methods.

`ManifoldsBase.shortest_geodesic`

— Method`shortest_geodesic(M::Manifold, p, q) -> Function`

Get a `geodesic`

$\gamma_{p,q}(t)$ whose length is the shortest path between the points `p`

and `q`

, where $\gamma_{p,q}(0)=p$ and $\gamma_{p,q}(1)=q$. When there are multiple shortest geodesics, there is no guarantee which will be returned.

This function returns a function of time, which may be a `Real`

or an `AbstractVector`

.

```
shortest_geodesic(M::Manifold, p, q, t::Real)
shortest_geodesic(M::Manifold, p, q, T::AbstractVector) -> AbstractVector
```

Return the point at time `t`

or points at times `t`

in `T`

along the shortest geodesic.

`ManifoldsBase.zero_tangent_vector!`

— Method`zero_tangent_vector!(M::Manifold, X, p)`

Save to `X`

a vector such that retracting `X`

to the `Manifold`

`M`

at `p`

produces `p`

.

`ManifoldsBase.zero_tangent_vector`

— Method`zero_tangent_vector(M::Manifold, p)`

Return the tangent vector from the tangent space at `p`

on the `Manifold`

`M`

, that represents the zero vector, i.e. such that a retraction at `p`

produces `p`

.

## Number systems

`ManifoldsBase.AbstractNumbers`

— Type`AbstractNumbers`

An abstract type to represent the number system on which a manifold is built.

This provides concrete number types for dispatch. The two most common number types are the fields `RealNumbers`

(`ℝ`

for short) and `ComplexNumbers`

(`ℂ`

).

`ManifoldsBase.ComplexNumbers`

— Type```
ComplexNumbers <: AbstractNumbers
ℂ = ComplexNumbers()
```

The field of complex numbers.

`ManifoldsBase.QuaternionNumbers`

— Type```
QuaternionNumbers <: AbstractNumbers
ℍ = QuaternionNumbers()
```

The division algebra of quaternions.

`ManifoldsBase.RealNumbers`

— Type```
RealNumbers <: AbstractNumbers
ℝ = RealNumbers()
```

The field of real numbers.

`ManifoldsBase._unify_number_systems`

— Method`_unify_number_systems(𝔽s::AbstractNumbers...)`

Compute a number system that includes all given number systems (as sub-systems) and is closed under addition and multiplication.

`ManifoldsBase.number_system`

— Method`number_system(M::Manifold{𝔽})`

Return the number system the manifold `M`

is based on, i.e. the parameter `𝔽`

.

`ManifoldsBase.real_dimension`

— Method`real_dimension(𝔽::AbstractNumbers)`

Return the real dimension $\dim_ℝ 𝔽$ of the `AbstractNumbers`

system `𝔽`

. The real dimension is the dimension of a real vector space with which a number in `𝔽`

can be identified. For example, `ComplexNumbers`

have a real dimension of 2, and `QuaternionNumbers`

have a real dimension of 4.

## Allocation

Non-mutating functions in `ManifoldsBase.jl`

are typically implemented using mutating variants. Allocation of new points is performed using a custom mechanism that relies on the following functions:

`allocate`

that allocates a new point or vector similar to the given one. This function behaves like`similar`

for simple representations of points and vectors (for example`Array{Float64}`

). For more complex types, such as nested representations of`PowerManifold`

(see`NestedPowerRepresentation`

),`FVector`

types, checked types like`ValidationMPoint`

and more it operates differently. While`similar`

only concerns itself with the higher level of nested structures,`allocate`

maps itself through all levels of nesting until a simple array of numbers is reached and then calls`similar`

. The difference can be most easily seen in the following example:

```
julia> x = similar([[1.0], [2.0]])
2-element Array{Array{Float64,1},1}:
#undef
#undef
julia> y = Manifolds.allocate([[1.0], [2.0]])
2-element Array{Array{Float64,1},1}:
[6.90031725726027e-310]
[6.9003678131654e-310]
julia> x[1]
ERROR: UndefRefError: access to undefined reference
Stacktrace:
[1] getindex(::Array{Array{Float64,1},1}, ::Int64) at ./array.jl:744
[2] top-level scope at REPL[12]:1
julia> y[1]
1-element Array{Float64,1}:
6.90031725726027e-310
```

`allocate_result`

allocates a result of a particular function (for example [`exp`

], [`flat`

], etc.) on a particular manifold with particular arguments. It takes into account the possibility that different arguments may have different numeric`number_eltype`

types thorough the`ManifoldsBase.allocate_result_type`

function.

## Bases

The following functions and types provide support for bases of the tangent space of different manifolds. An orthonormal basis of the tangent space $T_p \mathcal M$ of (real) dimension $n$ has a real-coefficient basis $e_1, e_2, …, e_n$ if $\mathrm{Re}(g_p(e_i, e_j)) = δ_{ij}$ for each $i,j ∈ \{1, 2, …, n\}$ where $g_p$ is the Riemannian metric at point $p$. A vector $X$ from the tangent space $T_p \mathcal M$ can be expressed in Einstein notation as a sum $X = X^i e_i$, where (real) coefficients $X^i$ are calculated as $X^i = \mathrm{Re}(g_p(X, e_i))$.

The main types are:

`DefaultOrthonormalBasis`

, which is designed to work when no special properties of the tangent space basis are required. It is designed to make`get_coordinates`

and`get_vector`

fast.`DiagonalizingOrthonormalBasis`

, which diagonalizes the curvature tensor and makes the curvature in the selected direction equal to 0.`ProjectedOrthonormalBasis`

, which projects a basis of the ambient space and orthonormalizes projections to obtain a basis in a generic way.`CachedBasis`

, which stores (explicitly or implicitly) a precomputed basis at a certain point.

The main functions are:

`get_basis`

precomputes a basis at a certain point.`get_coordinates`

returns coordinates of a tangent vector.`get_vector`

returns a vector for the specified coordinates.`get_vectors`

returns a vector of basis vectors. Calling it should be avoided for high-dimensional manifolds.

`ManifoldsBase.AbstractBasis`

— Type`AbstractBasis{𝔽}`

Abstract type that represents a basis on a manifold or a subset of it.

The type parameter `𝔽`

denotes the `AbstractNumbers`

that will be used for the vectors elements.

`ManifoldsBase.AbstractOrthogonalBasis`

— Type`AbstractOrthogonalBasis{𝔽}`

Abstract type that represents an orthonormal basis on a manifold or a subset of it.

The type parameter `𝔽`

denotes the `AbstractNumbers`

that will be used for the vectors elements.

`ManifoldsBase.AbstractOrthonormalBasis`

— Type`AbstractOrthonormalBasis{𝔽}`

Abstract type that represents an orthonormal basis on a manifold or a subset of it.

The type parameter `𝔽`

denotes the `AbstractNumbers`

that will be used for the vectors elements.

`ManifoldsBase.CachedBasis`

— Type`CachedBasis{𝔽,V,<:AbstractBasis{𝔽}} <: AbstractBasis{𝔽}`

A cached version of the given `basis`

with precomputed basis vectors. The basis vectors are stored in `data`

, either explicitly (like in cached variants of `ProjectedOrthonormalBasis`

) or implicitly.

**Constructor**

`CachedBasis(basis::AbstractBasis, data)`

`ManifoldsBase.DefaultBasis`

— Type`DefaultBasis{𝔽}`

An arbitrary basis on a manifold. This will usually be the fastest basis available for a manifold.

The type parameter `𝔽`

denotes the `AbstractNumbers`

that will be used for the vectors elements

`ManifoldsBase.DefaultOrthogonalBasis`

— Type`DefaultOrthogonalBasis{𝔽}`

An arbitrary orthogonal basis on a manifold. This will usually be the fastest orthogonal basis available for a manifold.

The type parameter `𝔽`

denotes the `AbstractNumbers`

that will be used for the vectors elements.

`ManifoldsBase.DefaultOrthonormalBasis`

— Type`DefaultOrthonormalBasis(𝔽::AbstractNumbers = ℝ)`

An arbitrary orthonormal basis on a manifold. This will usually be the fastest orthonormal basis available for a manifold.

The type parameter `𝔽`

denotes the `AbstractNumbers`

that will be used for the vectors elements.

`ManifoldsBase.DiagonalizingOrthonormalBasis`

— Type`DiagonalizingOrthonormalBasis{𝔽,TV} <: AbstractOrthonormalBasis{𝔽}`

An orthonormal basis `Ξ`

as a vector of tangent vectors (of length determined by `manifold_dimension`

) in the tangent space that diagonalizes the curvature tensor $R(u,v)w$ and where the direction `frame_direction`

$v$ has curvature `0`

.

The type parameter `𝔽`

denotes the `AbstractNumbers`

that will be used for the vectors elements.

**Constructor**

`DiagonalizingOrthonormalBasis(frame_direction, 𝔽::AbstractNumbers = ℝ)`

`ManifoldsBase.ProjectedOrthonormalBasis`

— Type`ProjectedOrthonormalBasis(method::Symbol, 𝔽::AbstractNumbers = ℝ)`

An orthonormal basis that comes from orthonormalization of basis vectors of the ambient space projected onto the subspace representing the tangent space at a given point.

The type parameter `𝔽`

denotes the `AbstractNumbers`

that will be used for the vectors elements.

Available methods:

`:gram_schmidt`

uses a modified Gram-Schmidt orthonormalization.`:svd`

uses SVD decomposition to orthogonalize projected vectors. The SVD-based method should be more numerically stable at the cost of an additional assumption (local metric tensor at a point where the basis is calculated has to be diagonal).

`ManifoldsBase.allocation_promotion_function`

— Method`allocation_promotion_function(M::Manifold, f, args::Tuple)`

Determine the function that must be used to ensure that the allocated representation is of the right type. This is needed for `get_vector`

when a point on a complex manifold is represented by a real-valued vectors with a real-coefficient basis, so that a complex-valued vector representation is allocated.

`ManifoldsBase.get_basis`

— Method`get_basis(M::Manifold, p, B::AbstractBasis) -> CachedBasis`

Compute the basis vectors of the tangent space at a point on manifold `M`

represented by `p`

.

Returned object derives from `AbstractBasis`

and may have a field `.vectors`

that stores tangent vectors or it may store them implicitly, in which case the function `get_vectors`

needs to be used to retrieve the basis vectors.

See also: `get_coordinates`

, `get_vector`

`ManifoldsBase.get_coordinates`

— Method```
get_coordinates(M::Manifold, p, X, B::AbstractBasis)
get_coordinates(M::Manifold, p, X, B::CachedBasis)
```

Compute a one-dimensional vector of coefficients of the tangent vector `X`

at point denoted by `p`

on manifold `M`

in basis `B`

.

Depending on the basis, `p`

may not directly represent a point on the manifold. For example if a basis transported along a curve is used, `p`

may be the coordinate along the curve. If a `CachedBasis`

is provided, their stored vectors are used, otherwise the user has to provide a method to compute the coordinates.

For the `CachedBasis`

keep in mind that the reconstruction with `get_vector`

requires either a dual basis or the cached basis to be selfdual, for example orthonormal

See also: `get_vector`

, `get_basis`

`ManifoldsBase.get_vector`

— Method`get_vector(M::Manifold, p, X, B::AbstractBasis)`

Convert a one-dimensional vector of coefficients in a basis `B`

of the tangent space at `p`

on manifold `M`

to a tangent vector `X`

at `p`

.

Depending on the basis, `p`

may not directly represent a point on the manifold. For example if a basis transported along a curve is used, `p`

may be the coordinate along the curve.

For the `CachedBasis`

keep in mind that the reconstruction from `get_coordinates`

requires either a dual basis or the cached basis to be selfdual, for example orthonormal

See also: `get_coordinates`

, `get_basis`

`ManifoldsBase.get_vectors`

— Method`get_vectors(M::Manifold, p, B::AbstractBasis)`

Get the basis vectors of basis `B`

of the tangent space at point `p`

.

`ManifoldsBase.hat`

— Method`hat(M::Manifold, p, Xⁱ)`

Given a basis $e_i$ on the tangent space at a point `p`

and tangent component vector $X^i$, compute the equivalent vector representation $X=X^i e_i$, where Einstein summation notation is used:

For array manifolds, this converts a vector representation of the tangent vector to an array representation. The `vee`

map is the `hat`

map's inverse.

`ManifoldsBase.number_of_coordinates`

— Method`number_of_coordinates(M::Manifold, B::AbstractBasis)`

Compute the number of coordinates in basis `B`

of manifold `M`

. This also corresponds to the number of vectors represented by `B`

, or stored within `B`

in case of a `CachedBasis`

.

`ManifoldsBase.number_system`

— Method`number_system(::AbstractBasis)`

The number system for the vectors of the given basis.

`ManifoldsBase.vee`

— Method`vee(M::Manifold, p, X)`

Given a basis $e_i$ on the tangent space at a point `p`

and tangent vector `X`

, compute the vector components $X^i$, such that $X = X^i e_i$, where Einstein summation notation is used:

For array manifolds, this converts an array representation of the tangent vector to a vector representation. The `hat`

map is the `vee`

map's inverse.

## Vector transport

There are three main functions for vector transport:

Different types of vector transport are implemented using subtypes of `AbstractVectorTransportMethod`

:

`ManifoldsBase.AbstractVectorTransportMethod`

— Type`AbstractVectorTransportMethod`

Abstract type for methods for transporting vectors.

`ManifoldsBase.ParallelTransport`

— Type`ParallelTransport <: AbstractVectorTransportMethod`

Specify to use parallel transport as vector transport method within `vector_transport_to`

, `vector_transport_direction`

, or `vector_transport_along`

.

`ManifoldsBase.PoleLadderTransport`

— Type`PoleLadderTransport <: AbstractVectorTransportMethod`

Specify to use `pole_ladder`

as vector transport method within `vector_transport_to`

, `vector_transport_direction`

, or `vector_transport_along`

, i.e.

Let $X\in T_p\mathcal M$ be a tangent vector at $p\in\mathcal M$ and $q\in\mathcal M$ the point to transport to. Then $x = \exp_pX$ is used to call `y =`

`pole_ladder`

`(M, p, x, q)`

and the resulting vector is obtained by computing $Y = -\log_qy$.

The `PoleLadderTransport`

posesses two advantages compared to `SchildsLadderTransport`

:

- it is cheaper to evaluate, if you want to transport several vectors, since the mid point $c$ then stays unchanged.
- while both methods are exact if the curvature is zero, pole ladder is even exact in symmetric Riemannian manifolds
^{[Pennec2018]}

The pole ladder was was proposed in ^{[LorenziPennec2014]}. Its name stems from the fact that it resembles a pole ladder when applied to a sequence of points usccessively.

**Constructor**

```
PoleLadderTransport(
retraction = ExponentialRetraction(),
inverse_retraction = LogarithmicInverseRetraction(),
)
```

Construct the classical pole ladder that employs exp and log, i.e. as proposed in^{[LorenziPennec2014]}. For an even cheaper transport the inner operations can be changed to an `AbstractRetractionMethod`

`retraction`

and an `AbstractInverseRetractionMethod`

`inverse_retraction`

, respectively.

`ManifoldsBase.ProjectionTransport`

— Type`ProjectionTransport <: AbstractVectorTransportMethod`

Specify to use projection onto tangent space as vector transport method within `vector_transport_to`

, `vector_transport_direction`

, or `vector_transport_along`

. See `project`

for details.

`ManifoldsBase.SchildsLadderTransport`

— Type`SchildsLadderTransport <: AbstractVectorTransportMethod`

Specify to use `schilds_ladder`

as vector transport method within `vector_transport_to`

, `vector_transport_direction`

, or `vector_transport_along`

, i.e.

Let $X\in T_p\mathcal M$ be a tangent vector at $p\in\mathcal M$ and $q\in\mathcal M$ the point to transport to. Then

where $c$ is the mid point between $q$ and $d=\exp_pX$.

This method employs the internal function `schilds_ladder`

`(M, p, d, q)`

that avoids leaving the manifold.

The name stems from the image of this paralleltogram in a repeated application yielding the image of a ladder. The approximation was proposed in ^{[EhlersPiraniSchild1972]}.

**Constructor**

```
SchildsLadderTransport(
retraction = ExponentialRetraction(),
inverse_retraction = LogarithmicInverseRetraction(),
)
```

Construct the classical Schilds ladder that employs exp and log, i.e. as proposed in^{[EhlersPiraniSchild1972]}. For an even cheaper transport these inner operations can be changed to an `AbstractRetractionMethod`

`retraction`

and an `AbstractInverseRetractionMethod`

`inverse_retraction`

, respectively.

`ManifoldsBase.pole_ladder`

— Function```
pole_ladder(
M,
p,
d,
q,
c = mid_point(M, p, q);
retraction=ExponentialRetraction(),
inverse_retraction=LogarithmicInverseRetraction()
)
```

Compute an inner step of the pole ladder, that can be used as a `vector_transport_to`

. Let $c = \gamma_{p,q}(\frac{1}{2})$ mid point between `p`

and `q`

, then the pole ladder is given by

Where the classical pole ladder employs $\operatorname{retr}_d=\exp_d$ and $\operatorname{retr}_d^{-1}=\log_d$ but for an even cheaper transport these can be set to different `AbstractRetractionMethod`

and `AbstractInverseRetractionMethod`

.

When you have $X=log_pd$ and $Y = -\log_q \operatorname{Pl}(p,d,q)$, you will obtain the `PoleLadderTransport`

. When performing multiple steps, this method avoidsd the switching to the tangent space. Keep in mind that after $n$ successive steps the tangent vector reads $Y_n = (-1)^n\log_q \operatorname{Pl}(p_{n-1},d_{n-1},p_n)$.

It is cheaper to evaluate than `schilds_ladder`

, sinc if you want to form multiple ladder steps between `p`

and `q`

, but with different `d`

, there is just one evaluation of a geodesic each., since the center `c`

can be reused.

`ManifoldsBase.pole_ladder!`

— Function```
pole_ladder(
M,
pl,
p,
d,
q,
c = mid_point(M, p, q),
X = allocate_result_type(M, log, d, c);
retraction = ExponentialRetraction(),
inverse_retraction = LogarithmicInverseRetraction()
)
```

Compute the `pole_ladder`

, i.e. the result is saved in `pl`

. `X`

is used for storing intermediate inverse retraction.

`ManifoldsBase.schilds_ladder`

— Function```
schilds_ladder(
M,
p,
d,
q,
c = mid_point(M, q, d);
retraction = ExponentialRetraction(),
inverse_retraction = LogarithmicInverseRetraction(),
)
```

Perform an inner step of schilds ladder, which can be used as a `vector_transport_to`

, see `SchildsLadderTransport`

. Let $c = \gamma_{q,d}(\frac{1}{2})$ denote the mid point on the shortest geodesic connecting $q$ and the point $d$. Then Schild's ladder reads as

Where the classical Schilds ladder employs $\operatorname{retr}_d=\exp_d$ and $\operatorname{retr}_d^{-1}=\log_d$ but for an even cheaper transport these can be set to different `AbstractRetractionMethod`

and `AbstractInverseRetractionMethod`

.

In consistency with `pole_ladder`

you can change the way the mid point is computed using the optional parameter `c`

, but note that here it's the mid point between `q`

and `d`

.

When you have $X=log_pd$ and $Y = \log_q \operatorname{Sl}(p,d,q)$, you will obtain the `PoleLadderTransport`

. Then the approximation to the transported vector is given by $\log_q\operatorname{Sl}(p,d,q)$.

When performing multiple steps, this method avoidsd the switching to the tangent space. Hence after $n$ successive steps the tangent vector reads $Y_n = \log_q \operatorname{Pl}(p_{n-1},d_{n-1},p_n)$.

`ManifoldsBase.schilds_ladder!`

— Function```
schilds_ladder!(
M,
sl
p,
d,
q,
c = mid_point(M, q, d),
X = allocate_result_type(M, log, d, c);
retraction = ExponentialRetraction(),
inverse_retraction = LogarithmicInverseRetraction()
)
```

Compute `schilds_ladder`

and return the value in the parameter `sl`

. If the required mid point `c`

was computed before, it can be passed using `c`

, and the allocation of new memory can be avoided providing a tangent vector `X`

for the interims result.

`ManifoldsBase.vector_transport_along!`

— Method```
vector_transport_along!(
M::Manifold,
Y,
p,
X,
c::AbstractVector,
method::AbstractVectorTransportMethod
) where {T}
```

Compute the vector transport along a discretized curve `c`

using an `AbstractVectorTransportMethod`

`method`

succesively along the sampled curve.

`ManifoldsBase.vector_transport_along!`

— Method```
function vector_transport_along!(
M::Manifold,
Y,
p,
X,
c::AbstractVector,
method::PoleLadderTransport
)
```

Compute the vector transport along a discretized curve using `PoleLadderTransport`

succesively along the sampled curve. This method is avoiding additional allocations as well as inner exp/log by performing all ladder steps on the manifold and only computing one tangent vector in the end.

`ManifoldsBase.vector_transport_along!`

— Method```
vector_transport_along!(
M::Manifold,
Y,
p,
X,
c::AbstractVector,
method::SchildsLadderTransport
)
```

Compute the vector transport along a discretized curve using `SchildsLadderTransport`

succesively along the sampled curve. This method is avoiding additional allocations as well as inner exp/log by performing all ladder steps on the manifold and only computing one tangent vector in the end.

`ManifoldsBase.vector_transport_along!`

— Method```
vector_transport_along!(M::Manifold, Y, p, X, c)
vector_transport_along!(M::Manifold, Y, p, X, c, method::AbstractVectorTransportMethod)
```

Transport a vector `X`

from the tangent space at a point `p`

on the `Manifold`

`M`

along the curve represented by `c`

using the `method`

, which defaults to `ParallelTransport`

. The result is saved to `Y`

.

`ManifoldsBase.vector_transport_along`

— Method```
vector_transport_along(M::Manifold, p, X, c)
vector_transport_along(M::Manifold, p, X, c, method::AbstractVectorTransportMethod)
```

Transport a vector `X`

from the tangent space at a point `p`

on the `Manifold`

`M`

along the curve represented by `c`

using the `method`

, which defaults to `ParallelTransport`

.

`ManifoldsBase.vector_transport_direction!`

— Method```
vector_transport_direction!(M::Manifold, Y, p, X, d)
vector_transport_direction!(M::Manifold, Y, p, X, d, method::AbstractVectorTransportMethod)
```

Transport a vector `X`

from the tangent space at a point `p`

on the `Manifold`

`M`

in the direction indicated by the tangent vector `d`

at `p`

. By default, `exp`

and `vector_transport_to!`

are used with the `method`

, which defaults to `ParallelTransport`

. The result is saved to `Y`

.

`ManifoldsBase.vector_transport_direction`

— Method```
vector_transport_direction(M::Manifold, p, X, d)
vector_transport_direction(M::Manifold, p, X, d, method::AbstractVectorTransportMethod)
```

Transport a vector `X`

from the tangent space at a point `p`

on the `Manifold`

`M`

in the direction indicated by the tangent vector `d`

at `p`

. By default, `exp`

and `vector_transport_to!`

are used with the `method`

, which defaults to `ParallelTransport`

.

`ManifoldsBase.vector_transport_to!`

— Method`vector_transport_to!(M::Manifold, Y, p, X, q, method::PoleLadderTransport)`

Perform a vector transport by using `PoleLadderTransport`

.

`ManifoldsBase.vector_transport_to!`

— Method`vector_transport_to!(M::Manifold, Y, p, X, q, method::ProjectionTransport)`

Transport a vector `X`

from the tangent space at `p`

on the `Manifold`

`M`

by interpreting it as an element of the embedding and then projecting it onto the tangent space at `q`

. This function needs to be separately implemented for each manifold because projection `project`

may also change vector representation (if it's different than in the embedding) and it is assumed that the vector `X`

already has the correct representation for `M`

.

`ManifoldsBase.vector_transport_to!`

— Method`vector_transport_to!(M::Manifold, Y, p, X, q, method::SchildsLadderTransport)`

Perform a vector transport by using `SchildsLadderTransport`

.

`ManifoldsBase.vector_transport_to!`

— Method```
vector_transport_to!(M::Manifold, Y, p, X, q)
vector_transport_to!(M::Manifold, Y, p, X, q, method::AbstractVectorTransportMethod)
```

Transport a vector `X`

from the tangent space at a point `p`

on the `Manifold`

`M`

along the `shortest_geodesic`

to the tangent space at another point `q`

. By default, the `AbstractVectorTransportMethod`

`method`

is `ParallelTransport`

. The result is saved to `Y`

.

`ManifoldsBase.vector_transport_to`

— Method```
vector_transport_to(M::Manifold, p, X, q)
vector_transport_to(M::Manifold, p, X, q, method::AbstractVectorTransportMethod)
```

Transport a vector `X`

from the tangent space at a point `p`

on the `Manifold`

`M`

along the `shortest_geodesic`

to the tangent space at another point `q`

. By default, the `AbstractVectorTransportMethod`

`method`

is `ParallelTransport`

.

## A Decorator for manifolds

A decorator manifold extends the functionality of a `Manifold`

in a semi-transparent way. It internally stores the `Manifold`

it extends and by default for functions defined in the `ManifoldsBase`

it acts transparently in the sense that it passes all functions through to the base except those that it actually affects. For example, because the `ValidationManifold`

affects nearly all functions, it overwrites nearly all functions, except a few like `manifold_dimension`

. On the other hand, the `MetricManifold`

only affects functions that involve metrics, especially `exp`

and `log`

but not the `manifold_dimension`

. Contrary to the previous decorator, the `MetricManifold`

does not overwrite functions. The decorator sets functions like `exp`

and `log`

to be implemented anew but required to be implemented when specifying a new metric. An exception is not issued if a metric is additionally set to be the default metric (see `is_default_metric`

, since this makes all functions act transparently. this last case assumes that the newly specified metric type is actually the one already implemented on a manifold initially.

By default, i.e. for a plain new decorator, all functions are passed down. To implement a method for a decorator that behaves differently from the method of the same function for the internal manifold, two steps are required. Let's assume the function is called `f(M, arg1, arg2)`

, and our decorator manifold `DM`

of type `OurDecoratorManifold`

decorates `M`

. Then

- set
`decorator_transparent_dispatch(f, M::OurDecoratorManifold, args...) = Val(:intransparent)`

- implement
`f(DM::OurDecoratorManifold, arg1, arg2)`

This makes it possible to extend a manifold or all manifolds with a feature or replace a feature of the original manifold. The `MetricManifold`

is the best example of the second case, since the default metric indicates for which metric the manifold was originally implemented, such that those functions are just passed through. This can best be seen in the `SymmetricPositiveDefinite`

manifold with its `LinearAffineMetric`

.

`ManifoldsBase.@decorator_transparent_fallback`

— Macro```
@decorator_transparent_fallback(ex)
@decorator_transparent_fallback(fallback_case = :intransparent, ex)
```

This macro introduces an additional implementation for a certain additional case. This can especially be used if for an already transparent function and an abstract intermediate type a change in the default is required. For implementing a concrete type, neither this nor any other trick is necessary. One just implements the function as before. Note that a decorator that `is_default_decorator`

still dispatches to the transparent case.

`:transparent`

states, that the function is transparently passed on to the manifold that is decorated by the`AbstractDecoratorManifold`

`M`

, which is determined using the function`decorated_manifold`

.`:intransparent`

states that an implementation for this decorator is required, and if none of the types provides one, an error is issued. Since this macro provides such an implementation, this is the default.`:parent`

states, that this function passes on to the supertype instead of to the decorated manifold.

Inline definitions are not supported. The function signature however may contain keyword arguments and a where clause. It does not allow for parameters with default values.

**Examples**

```
@decorator_transparent_fallback function log!(M::AbstractGroupManifold, X, p, q)
log!(decorated_manifold(M), X, p, Q)
end
@decorator_transparent_fallback :transparent function log!(M::AbstractGroupManifold, X, p, q)
log!(decorated_manifold(M), X, p, Q)
end
```

`ManifoldsBase.@decorator_transparent_function`

— Macro```
@decorator_transparent_function(ex)
@decorator_transparent_function(fallback_case = :intransparent, ex)
```

Introduce the function specified by `ex`

to act transparently with respect to `AbstractDecoratorManifold`

s. This introduces the possibility to modify the kind of transparency the implementation is done for. This optional first argument, the `Symbol`

within `fallback_case`

. This macro can be used to define a function and introduce it as transparent to other decorators. Note that a decorator that `is_default_decorator`

still dispatches to the transparent case.

The cases of transparency are

`:transparent`

states, that the function is transparently passed on to the manifold that is decorated by the`AbstractDecoratorManifold`

`M`

, which is determined using the function`decorated_manifold`

.`:intransparent`

states that an implementation for this decorator is required, and if none of the types provides one, an error is issued. Since this macro provides such an implementation, this is the default.`:parent`

states, that this function passes on to the supertype instead of to the decorated manifold. Passing is performed using the`invoke`

function where the type of manifold is replaced by its supertype.

Innkoline-definitions are not yet covered – the function signature however may contain keyword arguments and a where clause.

**Examples**

```
@decorator_transparent_function log!(M::AbstractDecoratorManifold, X, p, q)
log!(decorated_manifold(M), X, p, Q)
end
@decorator_transparent_function :parent log!(M::AbstractDecoratorManifold, X, p, q)
log!(decorated_manifold(M), X, p, Q)
end
```

`ManifoldsBase.@decorator_transparent_signature`

— Macro`@decorator_transparent_signature(ex)`

Introduces a given function to be transparent with respect to all decorators. The function is adressed by its signature in `ex`

.

Supports standard, keyword arguments and `where`

clauses. Doesn't support parameters with default values. It introduces a dispatch on several transparency modes

The cases of transparency are

`:transparent`

states, that the function is transparently passed on to the manifold that is decorated by the`AbstractDecoratorManifold`

`M`

, which is determined using the function`decorated_manifold`

. This is the default.`:intransparent`

states that an implementation for this decorator is required, and if none of the types provides one, an error is issued.`:parent`

states, that this function passes on to the supertype instead of to the decorated manifold.

Inline definitions are not supported. The function signature however may contain keyword arguments and a where clause.

The dispatch kind can later still be set to something different, see `decorator_transparent_dispatch`

**Examples:**

```
@decorator_transparent_signature log!(M::AbstractDecoratorManifold, X, p, q)
@decorator_transparent_signature log!(M::TD, X, p, q) where {TD<:AbstractDecoratorManifold}
@decorator_transparent_signature isapprox(M::AbstractDecoratorManifold, p, q; kwargs...)
```

`ManifoldsBase.AbstractDecoratorManifold`

— Type`AbstractDecoratorManifold{𝔽} <: Manifold{𝔽}`

An `AbstractDecoratorManifold`

indicates that to some extent a manifold subtype decorates another `Manifold`

in the sense that it either

- it extends the functionality of a manifold with further features
- it defines a new manifold that internally uses functions from the decorated manifold

with the main intent that several or most functions of `Manifold`

are transparently passed through to the manifold that is decorated. This way a function implemented for a decorator acts transparent on all other decorators, i.e. they just pass them through. If the decorator the function is implemented for is not among the decorators, an error is issued. By default all base manifold functions, for example `exp`

and `log`

are transparent for all decorators.

Transparency of functions with respect to decorators can be specified using the macros `@decorator_transparent_fallback`

, `@decorator_transparent_function`

and `@decorator_transparent_signature`

.

`ManifoldsBase.decorated_manifold`

— Method`decorated_manifold(M::AbstractDecoratorManifold)`

Return the manifold decorated by the decorator `M`

. Defaults to `M.manifold`

.

`ManifoldsBase.decorator_transparent_dispatch`

— Method`decorator_transparent_dispatch(f, M::Manifold, args...) -> Val`

Given a `Manifold`

`M`

and a function `f(M,args...)`

, indicate, whether a function is `Val(:transparent)`

or `Val(:intransparent)`

for the (decorated) `Manifold`

`M`

. Another possibility is, that for `M`

and given `args...`

the function `f`

should invoke `M`

s `Val(:parent)`

implementation, see `@decorator_transparent_function`

for details.

`ManifoldsBase.default_decorator_dispatch`

— Method`default_decorator_dispatch(M) -> Val`

Return whether by default to dispatch the the inner manifold of a decorator (`Val(true)`

) or not (`Val(false`

). For more details see `is_decorator_transparent`

.

`ManifoldsBase.is_decorator_transparent`

— Method`is_decorator_transparent(f, M::Manifold, args...) -> Bool`

Given a `Manifold`

`M`

and a function `f(M, args...)`

, indicate, whether an `AbstractDecoratorManifold`

acts transparently for `f`

. This means, it just passes through down to the internally stored manifold. Transparency is only defined for decorator manifolds and by default all decorators are transparent. A function that is affected by the decorator indicates this by returning `false`

. To change this behaviour, see `decorator_transparent_dispatch`

.

If a decorator manifold is not in general transparent, it might still pass down for the case that a decorator is the default decorator, see `is_default_decorator`

.

`ManifoldsBase.is_default_decorator`

— Method`is_default_decorator(M) -> Bool`

For any manifold that is a subtype of `AbstractDecoratorManifold`

, this function indicates whether a certain manifold `M`

acts as a default decorator.

This yields that *all* functions are passed through to the decorated `Manifold`

if `M`

is indicated as default. This overwrites all `is_decorator_transparent`

values.

This yields the following advantange: For a manifold one usually implicitly assumes for example a metric. To avoid reimplementation of this metric when introducing a second metric, the first metric can be set to be the default, i.e. its implementaion is already given by the undecorated case.

Value returned by this function is determined by `default_decorator_dispatch`

, which returns a `Val`

-wrapped boolean for type stability of certain functions.

## ValidationManifold

`ValidationManifold`

is a simple decorator that “decorates” a manifold with tests that all involved arrays are correct. For example involved input and output paratemers are checked before and after running a function, repectively. This is done by calling `is_manifold_point`

or `is_tangent_vector`

whenever applicable.

`ManifoldsBase.ValidationCoTVector`

— Type`ValidationCoTVector <: CoTVector`

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 and `ValidationTVector`

s.

`ManifoldsBase.ValidationMPoint`

— Type`ValidationMPoint <: MPoint`

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<:Manifold{𝔽}} <: AbstractDecoratorManifold{𝔽}`

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

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 `Manifold`

`M`

with types points, vectors, and covectors.

`ManifoldsBase.ValidationTVector`

— Type`ValidationTVector <: TVector`

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 and `ValidationCoTVector`

s.

`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.

## EmbeddedManifold

Some manifolds can easily be defined by using a certain embedding. For example the `Sphere`

`(n)`

is embedded in `Euclidean`

`(n+1)`

. Similar to the metric and `MetricManifold`

, an embedding is often implicitly assumed. We introduce the embedded manifolds hence as an `AbstractDecoratorManifold`

.

This decorator enables to use such an embedding in an transparent way. Different types of embeddings can be distinguished using the `AbstractEmbeddingType`

.

The embedding also covers representation of tangent vectors. For both points and tangent vectors the function `embed`

returns their representation in the embedding. For any point or vector in the embedding the functions `project`

can be used to obtain the closest point on the manifold and tangent vector in the tangent space. Note that both `embed`

and `project`

may also be implemented on a `Manifold`

already, assuming that the default implementation of that manifold has an (implicitly assumed) embedding. If there exists more than one embedding or to foster using functions from the embedding, like `inner`

, this decorator can be employed. The different embeddings might also refer to different representations or parametrisations of an embedding. A specific example where `embed`

and `project`

actually do change the representation are Lie groups. Here, tangent vectors are often represented in the Lie algebra, i.e. in the tangent space of the identity element. Then, projection first projects onto a tangent space but also takes care to return the correct representation, i.e. the element from the Lie algebra corresponding to the projected tangent vector. Similarly, before embedding a tangent vector that is represented by an element from the Lie algebra, the group opration is applied.

### Isometric Embeddings

For isometric embeddings the type `AbstractIsometricEmbeddingType`

can be used to avoid reimplementing the metric. See `Sphere`

or `Hyperbolic`

for example. Here, the exponential map, the logarithmic map, the retraction and its inverse are set to `:intransparent`

, i.e. they have to be implemented.

Furthermore, the `TransparentIsometricEmbedding`

type even states that the exponential and logarithmic maps as well as retractions and vector transports of the embedding can be used for the embedded manifold as well. See `SymmetricMatrices`

for an example.

In both cases of course `check_manifold_point`

and `check_tangent_vector`

have to be implemented.

### Technical Details

Semantically we use the idea of the embedding to efficiently implement a manifold by not having to implement those functions that are already given by its embedding. Hence we decorate in some sense the manifold we implement. Still, technically `base_manifold`

returns the embedding as long as `EmbeddedManifold`

is used. For the abstract case, `AbstractEmbeddedManifold`

the base manfiold might differ. Note that internally `base_manifold`

uses `decorated_manifold`

for one step of removing multiple decorators.

Clearly `get_embedding`

always returns the embedding.

### Types

`ManifoldsBase.AbstractEmbeddedManifold`

— Type`AbstractEmbeddedManifold{𝔽,T<:AbstractEmbeddingType,𝔽} <: AbstractDecoratorManifold{𝔽}`

An abstract type for embedded manifolds, which acts as an `AbstractDecoratorManifold`

. The functions of the manifold that is embedded can hence be just passed on to the embedding. The embedding is further specified by an `AbstractEmbeddingType`

.

This means, that technically an embedded manifold is a decorator for the embedding, i.e. functions of this type get, in the semi-transparent way of the `AbstractDecoratorManifold`

, passed on to the embedding.

`ManifoldsBase.AbstractEmbeddingType`

— Type`AbstractEmbeddingType`

A type used to specify properties of an `AbstractEmbeddedManifold`

.

`ManifoldsBase.AbstractIsometricEmbeddingType`

— Type`AbstractIsometricEmbeddingType <: AbstractEmbeddingType`

Characterizes an embedding as isometric. For this case the `inner`

product is passed from the embedded manifold to the embedding.

`ManifoldsBase.DefaultEmbeddingType`

— Type`DefaultEmbeddingType <: AbstractEmbeddingType`

A type of default embedding that does not have any special properties.

`ManifoldsBase.DefaultIsometricEmbeddingType`

— Type`DefaultIsometricEmbeddingType <: AbstractIsometricEmbeddingType`

An isometric embedding type that acts as a default, i.e. it has no specifig properties beyond its isometric property.

`ManifoldsBase.EmbeddedManifold`

— Type`EmbeddedManifold{𝔽, MT <: Manifold, NT <: Manifold, ET} <: AbstractEmbeddedManifold{𝔽, ET}`

A type to represent that a `Manifold`

`M`

of type `MT`

is indeed an emebedded manifold and embedded into the manifold `N`

of type `NT`

. Based on the `AbstractEmbeddingType`

`ET`

, this introduces methods for `M`

by passing them through to embedding `N`

.

**Fields**

`manifold`

the manifold that is an embedded manifold`embedding`

a second manifold, the first one is embedded into

**Constructor**

`EmbeddedManifold(M, N, e=TransparentIsometricEmbedding())`

Generate the `EmbeddedManifold`

of the `Manifold`

`M`

into the `Manifold`

`N`

with `AbstractEmbeddingType`

`e`

that by default is the most transparent `TransparentIsometricEmbedding`

`ManifoldsBase.TransparentIsometricEmbedding`

— Type`TransparentIsometricEmbedding <: AbstractIsometricEmbeddingType`

Specify that an embedding is the default isometric embedding. This even inherits logarithmic and exponential map as well as retraction and inverse retractions from the embedding.

For an example, see `SymmetricMatrices`

which are isometrically embedded in the Euclidean space of matrices but also inherit exponential and logarithmic maps.

### Functions

`ManifoldsBase.base_manifold`

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

Return the base manifold of `M`

that is enhanced with its embedding. While functions like `inner`

might be overwritten to use the (decorated) manifold representing the embedding, the base*manifold is the manifold itself in the sense that detemining e.g. the [`is*default_metric`](@ref) does not fall back to check with the embedding but with the manifold itself. For this abstract case, just`

M` is returned.

`ManifoldsBase.base_manifold`

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

Return the base manifold of `M`

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

is returned.

`ManifoldsBase.check_manifold_point`

— Method`check_manifold_point(M::AbstractEmbeddedManifold, p; kwargs)`

check whether a point `p`

is a valid point on the `AbstractEmbeddedManifold`

, i.e. that `embed(M, p)`

is a valid point on the embedded manifold.

`ManifoldsBase.check_tangent_vector`

— Method`check_tangent_vector(M::AbstractEmbeddedManifold, p, X; check_base_point = true, kwargs...)`

check that `embed(M,p,X)`

is a valid tangent to `embed(p,X)`

, where `check_base_point`

determines whether the validity of `p`

is checked, too.

`ManifoldsBase.default_embedding_dispatch`

— Method`default_embedding_dispatch(M::AbstractEmbeddedManifold)`

This method indicates that an `AbstractEmbeddedManifold`

is the default and hence acts completely transparently and passes all functions transparently onwards. This is used by the `AbstractDecoratorManifold`

within `default_decorator_dispatch`

. By default this is set to `Val(false)`

.

`ManifoldsBase.get_embedding`

— Method`get_embedding(M::AbstractEmbeddedManifold)`

Return the `Manifold`

`N`

an `AbstractEmbeddedManifold`

is embedded into.

## DefaultManifold

`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 <: Manifold`

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 variabes.

- LorenziPennec2014
Lorenzi, M. and Pennec, X: Efficient parallel transport of deformations in time series of images: From Schild’s to pole ladder. Journal of Mathematical Imaging and Vision (2014), 50(1), pp. 5–17 doi 10.1007/s10851-013-0470-3, hal: hal-00870489

- Pennec2018
Pennec, X: Parallel Transport with Pole Ladder: a Third Order Scheme in Affine Connection Spaces which is Exact in Affine Symmetric Spaces. arXiv: 1805.11436

- EhlersPiraniSchild1972
Ehlers, J., Pirani, F.A.E., Schild, A.: The geometry of free fall and light propagation. In: O’Raifeartaigh, L. (ed.) General Relativity: Papers in Honour of J. L. Synge, pp. 63–84. Clarendon Press, Oxford (1972). reprint doi: 10.1007/s10714-012-1353-4