# Group manifolds and actions

Lie groups, groups that are Riemannian manifolds with a smooth binary group operation AbstractGroupOperation, are implemented as AbstractDecoratorManifold and specifying the group operation using the IsGroupManifold or by decorating an existing manifold with a group operation using GroupManifold.

The common addition and multiplication group operations of AdditionOperation and MultiplicationOperation are provided, though their behavior may be customized for a specific group.

There are short introductions at the beginning of each subsection. They briefly mention what is available with links to more detailed descriptions.

## Groups

The following operations are available for group manifolds:

### Group manifold

GroupManifold adds a group structure to the wrapped manifold. It does not affect metric (or connection) structure of the wrapped manifold, however it can to be further wrapped in MetricManifold to get invariant metrics, or in a ConnectionManifold to equip it with a Cartan-Schouten connection.

Manifolds.AbstractGroupOperationType
AbstractGroupOperation

Abstract type for smooth binary operations $∘$ on elements of a Lie group $\mathcal{G}$:

$$$∘ : \mathcal{G} × \mathcal{G} → \mathcal{G}$$$

An operation can be either defined for a specific group manifold over number system 𝔽 or in general, by defining for an operation Op the following methods:

identity_element!(::AbstractDecoratorManifold, q, q)
inv!(::AbstractDecoratorManifold, q, p)
_compose!(::AbstractDecoratorManifold, x, p, q)

Note that a manifold is connected with an operation by wrapping it with a decorator, AbstractDecoratorManifold using the IsGroupManifold to specify the operation. For a concrete case the concrete wrapper GroupManifold can be used.

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Manifolds.IdentityType
Identity{O<:AbstractGroupOperation}

Represent the group identity element $e ∈ \mathcal{G}$ on a Lie group $\mathcal G$ with AbstractGroupOperation of type O.

Similar to the philosophy that points are agnostic of their group at hand, the identity does not store the group g it belongs to. However it depends on the type of the AbstractGroupOperation used.

See also identity_element on how to obtain the corresponding AbstractManifoldPoint or array representation.

Constructors

Identity(G::AbstractDecoratorManifold{𝔽})
Identity(o::O)
Identity(::Type{O})

create the identity of the corresponding subtype O<:AbstractGroupOperation

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Manifolds.IsGroupManifoldType
IsGroupManifold{O<:AbstractGroupOperation} <: AbstractTrait

A trait to declare an AbstractManifold as a manifold with group structure with operation of type O.

Using this trait you can turn a manifold that you implement implictly into a Lie group. If you wish to decorate an existing manifold with one (or different) AbstractGroupActions, see GroupManifold.

Constructor

IsGroupManifold(op)
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Base.invMethod
inv(G::AbstractDecoratorManifold, p)

Inverse $p^{-1} ∈ \mathcal{G}$ of an element $p ∈ \mathcal{G}$, such that $p \circ p^{-1} = p^{-1} \circ p = e ∈ \mathcal{G}$, where $e$ is the Identity element of $\mathcal{G}$.

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Manifolds.adjoint_actionMethod
adjoint_action(G::AbstractDecoratorManifold, p, X)

Adjoint action of the element p of the Lie group G on the element X of the corresponding Lie algebra.

It is defined as the differential of the group authomorphism $Ψ_p(q) = pqp⁻¹$ at the identity of G.

$$$\operatorname{Ad}_p(X) = dΨ_p(e)[X]$$$

where $e$ is the identity element of G.

Note that the adjoint representation of a Lie group isn't generally faithful. Notably the adjoint representation of SO(2) is trivial.

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Manifolds.composeMethod
compose(G::AbstractDecoratorManifold, p, q)

Compose elements $p,q ∈ \mathcal{G}$ using the group operation $p \circ q$.

For implementing composition on a new group manifold, please overload _compose instead so that methods with Identity arguments are not ambiguous.

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Manifolds.exp_lieMethod
exp_lie(G, X)
exp_lie!(G, q, X)

Compute the group exponential of the Lie algebra element X. It is equivalent to the exponential map defined by the CartanSchoutenMinus connection.

Given an element $X ∈ 𝔤 = T_e \mathcal{G}$, where $e$ is the Identity element of the group $\mathcal{G}$, and $𝔤$ is its Lie algebra, the group exponential is the map

$$$\exp : 𝔤 → \mathcal{G},$$$

such that for $t,s ∈ ℝ$, $γ(t) = \exp (t X)$ defines a one-parameter subgroup with the following properties. Note that one-parameter subgroups are commutative (see [Suhubi2013], section 3.5), even if the Lie group itself is not commutative.

\begin{aligned} γ(t) &= γ(-t)^{-1}\\ γ(t + s) &= γ(t) \circ γ(s) = γ(s) \circ γ(t)\\ γ(0) &= e\\ \lim_{t → 0} \frac{d}{dt} γ(t) &= X. \end{aligned}
Note

In general, the group exponential map is distinct from the Riemannian exponential map exp.

For example for the MultiplicationOperation and either Number or AbstractMatrix the Lie exponential is the numeric/matrix exponential.

$$$\exp X = \operatorname{Exp} X = \sum_{n=0}^∞ \frac{1}{n!} X^n.$$$

Since this function also depends on the group operation, make sure to implement the corresponding trait version exp_lie(::TraitList{<:IsGroupManifold}, G, X).

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Manifolds.inverse_translateMethod
inverse_translate(G::AbstractDecoratorManifold, p, q, conv::ActionDirection=LeftAction())

Inverse translate group element $q$ by $p$ with the inverse translation $τ_p^{-1}$ with the specified convention, either left ($L_p^{-1}$) or right ($R_p^{-1}$), defined as

\begin{aligned} L_p^{-1} &: q ↦ p^{-1} \circ q\\ R_p^{-1} &: q ↦ q \circ p^{-1}. \end{aligned}
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Manifolds.inverse_translate_diffMethod
inverse_translate_diff(G::AbstractDecoratorManifold, p, q, X, conv::ActionDirection=LeftAction())

For group elements $p, q ∈ \mathcal{G}$ and tangent vector $X ∈ T_q \mathcal{G}$, compute the action on $X$ of the differential of the inverse translation $τ_p$ by $p$, with the specified left or right convention. The differential transports vectors:

$$$(\mathrm{d}τ_p^{-1})_q : T_q \mathcal{G} → T_{τ_p^{-1} q} \mathcal{G}\\$$$
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Manifolds.lie_bracketMethod
lie_bracket(G::AbstractDecoratorManifold, X, Y)

Lie bracket between elements X and Y of the Lie algebra corresponding to the Lie group G, cf. IsGroupManifold.

This can be used to compute the adjoint representation of a Lie algebra. Note that this representation isn't generally faithful. Notably the adjoint representation of 𝔰𝔬(2) is trivial.

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Manifolds.log_lieMethod
log_lie(G, q)
log_lie!(G, X, q)

Compute the Lie group logarithm of the Lie group element q. It is equivalent to the logarithmic map defined by the CartanSchoutenMinus connection.

Given an element $q ∈ \mathcal{G}$, compute the right inverse of the group exponential map exp_lie, that is, the element $\log q = X ∈ 𝔤 = T_e \mathcal{G}$, such that $q = \exp X$

Note

In general, the group logarithm map is distinct from the Riemannian logarithm map log.

For matrix Lie groups this is equal to the (matrix) logarithm:

$$$\log q = \operatorname{Log} q = \sum_{n=1}^∞ \frac{(-1)^{n+1}}{n} (q - e)^n,$$$

where $e$ here is the Identity element, that is, $1$ for numeric $q$ or the identity matrix $I_m$ for matrix $q ∈ ℝ^{m × m}$.

Since this function also depends on the group operation, make sure to implement either

• _log_lie(G, q) and _log_lie!(G, X, q) for the points not being the Identity
• the trait version log_lie(::TraitList{<:IsGroupManifold}, G, e), log_lie(::TraitList{<:IsGroupManifold}, G, X, e) for own implementations of the identity case.
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Manifolds.translateMethod
translate(G::AbstractDecoratorManifold, p, q, conv::ActionDirection=LeftAction()])

Translate group element $q$ by $p$ with the translation $τ_p$ with the specified convention, either left ($L_p$) or right ($R_p$), defined as

\begin{aligned} L_p &: q ↦ p \circ q\\ R_p &: q ↦ q \circ p. \end{aligned}
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Manifolds.translate_diffMethod
translate_diff(G::AbstractDecoratorManifold, p, q, X, conv::ActionDirection=LeftAction())

For group elements $p, q ∈ \mathcal{G}$ and tangent vector $X ∈ T_q \mathcal{G}$, compute the action of the differential of the translation $τ_p$ by $p$ on $X$, with the specified left or right convention. The differential transports vectors:

$$$(\mathrm{d}τ_p)_q : T_q \mathcal{G} → T_{τ_p q} \mathcal{G}\\$$$
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ManifoldsBase.hatMethod
hat(M::AbstractDecoratorManifold{𝔽,O}, ::Identity{O}, Xⁱ) where {𝔽,O<:AbstractGroupOperation}

Given a basis $e_i$ on the tangent space at a the Identity and tangent component vector $X^i$, compute the equivalent vector representation X=X^i e_i**, where Einstein summation notation is used:

$$$∧ : X^i ↦ X^i e_i$$$

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.

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ManifoldsBase.inverse_retractMethod
inverse_retract(
G::AbstractDecoratorManifold,
p,
X,
method::GroupLogarithmicInverseRetraction{<:ActionDirection},
)

Compute the inverse retraction using the group logarithm log_lie "translated" to any point on the manifold. With a group translation (translate) $τ_p$ in a specified direction, the retraction is

$$$\operatorname{retr}_p^{-1} = (\mathrm{d}τ_p)_e \circ \log \circ τ_p^{-1},$$$

where $\log$ is the group logarithm (log_lie), and $(\mathrm{d}τ_p)_e$ is the action of the differential of translation $τ_p$ evaluated at the identity element $e$ (see translate_diff).

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ManifoldsBase.retractMethod
retract(
G::AbstractDecoratorManifold,
p,
X,
method::GroupExponentialRetraction{<:ActionDirection},
)

Compute the retraction using the group exponential exp_lie "translated" to any point on the manifold. With a group translation (translate) $τ_p$ in a specified direction, the retraction is

$$$\operatorname{retr}_p = τ_p \circ \exp \circ (\mathrm{d}τ_p^{-1})_p,$$$

where $\exp$ is the group exponential (exp_lie), and $(\mathrm{d}τ_p^{-1})_p$ is the action of the differential of inverse translation $τ_p^{-1}$ evaluated at $p$ (see inverse_translate_diff).

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ManifoldsBase.veeMethod
vee(M::AbstractManifold, 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:

$$$\vee : X^i e_i ↦ X^i$$$

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.

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### GroupManifold

As a concrete wrapper for manifolds (e.g. when the manifold per se is a group manifold but another group structure should be implemented), there is the GroupManifold

Manifolds.GroupManifoldType
GroupManifold{𝔽,M<:AbstractManifold{𝔽},O<:AbstractGroupOperation} <: AbstractDecoratorManifold{𝔽}

Decorator for a smooth manifold that equips the manifold with a group operation, thus making it a Lie group. See IsGroupManifold for more details.

Group manifolds by default forward metric-related operations to the wrapped manifold.

Constructor

GroupManifold(manifold, op)
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### Generic Operations

For groups based on an addition operation or a group operation, several default implementations are provided.

### General linear group

Manifolds.GeneralLinearType
GeneralLinear{n,𝔽} <:
AbstractDecoratorManifold{𝔽}

The general linear group, that is, the group of all invertible matrices in $𝔽^{n×n}$.

The default metric is the left-$\mathrm{GL}(n)$-right-$\mathrm{O}(n)$-invariant metric whose inner product is

$$$⟨X_p,Y_p⟩_p = ⟨p^{-1}X_p,p^{-1}Y_p⟩_\mathrm{F} = ⟨X_e, Y_e⟩_\mathrm{F},$$$

where $X_p, Y_p ∈ T_p \mathrm{GL}(n, 𝔽)$, $X_e = p^{-1}X_p ∈ 𝔤𝔩(n) = T_e \mathrm{GL}(n, 𝔽) = 𝔽^{n×n}$ is the corresponding vector in the Lie algebra, and $⟨⋅,⋅⟩_\mathrm{F}$ denotes the Frobenius inner product.

By default, tangent vectors $X_p$ are represented with their corresponding Lie algebra vectors $X_e = p^{-1}X_p$.

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Base.logMethod
log(G::GeneralLinear, p, q)

Compute the logarithmic map on the GeneralLinear(n) group.

The algorithm proceeds in two stages. First, the point $r = p^{-1} q$ is projected to the nearest element (under the Frobenius norm) of the direct product subgroup $\mathrm{O}(n) × S^+$, whose logarithmic map is exactly computed using the matrix logarithm. This initial tangent vector is then refined using the NLSolveInverseRetraction.

For GeneralLinear(n, ℂ), the logarithmic map is instead computed on the realified supergroup GeneralLinear(2n) and the resulting tangent vector is then complexified.

Note that this implementation is experimental.

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### Heisenberg group

Manifolds.HeisenbergGroupType
HeisenbergGroup{n} <: AbstractDecoratorManifold{ℝ}

Heisenberg group HeisenbergGroup(n) is the group of $(n+2) × (n+2)$ matrices [BinzPods2008]

$$$\begin{bmatrix} 1 & \mathbf{a} & c \\ \mathbf{0} & I_n & \mathbf{b} \\ 0 & \mathbf{0} & 1 \end{bmatrix}$$$

where $I_n$ is the $n×n$ unit matrix, $\mathbf{a}$ is a row vector of length $n$, $\mathbf{b}$ is a column vector of length $n$ and $c$ is a real number. The group operation is matrix multiplication.

The left-invariant metric on the manifold is used.

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Base.expMethod
exp(M::HeisenbergGroup, p, X)

Exponential map on the HeisenbergGroup M with the left-invariant metric. The expression reads

$$$\exp_{\begin{bmatrix} 1 & \mathbf{a}_p & c_p \\ \mathbf{0} & I_n & \mathbf{b}_p \\ 0 & \mathbf{0} & 1 \end{bmatrix}}\left(\begin{bmatrix} 0 & \mathbf{a}_X & c_X \\ \mathbf{0} & 0_n & \mathbf{b}_X \\ 0 & \mathbf{0} & 0 \end{bmatrix}\right) = \begin{bmatrix} 1 & \mathbf{a}_p + \mathbf{a}_X & c_p + c_X + \mathbf{a}_X⋅\mathbf{b}_X/2 + \mathbf{a}_p⋅\mathbf{b}_X \\ \mathbf{0} & I_n & \mathbf{b}_p + \mathbf{b}_X \\ 0 & \mathbf{0} & 1 \end{bmatrix}$$$

where $I_n$ is the $n×n$ identity matrix, $0_n$ is the $n×n$ zero matrix and $\mathbf{a}⋅\mathbf{b}$ is dot product of vectors.

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Base.logMethod
log(G::HeisenbergGroup, p, q)

Compute the logarithmic map on the HeisenbergGroup group. The formula reads

$$$\log_{\begin{bmatrix} 1 & \mathbf{a}_p & c_p \\ \mathbf{0} & I_n & \mathbf{b}_p \\ 0 & \mathbf{0} & 1 \end{bmatrix}}\left(\begin{bmatrix} 1 & \mathbf{a}_q & c_q \\ \mathbf{0} & I_n & \mathbf{b}_q \\ 0 & \mathbf{0} & 1 \end{bmatrix}\right) = \begin{bmatrix} 0 & \mathbf{a}_q - \mathbf{a}_p & c_q - c_p + \mathbf{a}_p⋅\mathbf{b}_p - \mathbf{a}_q⋅\mathbf{b}_q - (\mathbf{a}_q - \mathbf{a}_p)⋅(\mathbf{b}_q - \mathbf{b}_p) / 2 \\ \mathbf{0} & 0_n & \mathbf{b}_q - \mathbf{b}_p \\ 0 & \mathbf{0} & 0 \end{bmatrix}$$$

where $I_n$ is the $n×n$ identity matrix, $0_n$ is the $n×n$ zero matrix and $\mathbf{a}⋅\mathbf{b}$ is dot product of vectors.

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Manifolds.exp_lieMethod
exp_lie(M::HeisenbergGroup, X)

Lie group exponential for the HeisenbergGroup M of the vector X. The formula reads

$$$\exp\left(\begin{bmatrix} 0 & \mathbf{a} & c \\ \mathbf{0} & 0_n & \mathbf{b} \\ 0 & \mathbf{0} & 0 \end{bmatrix}\right) = \begin{bmatrix} 1 & \mathbf{a} & c + \mathbf{a}⋅\mathbf{b}/2 \\ \mathbf{0} & I_n & \mathbf{b} \\ 0 & \mathbf{0} & 1 \end{bmatrix}$$$

where $I_n$ is the $n×n$ identity matrix, $0_n$ is the $n×n$ zero matrix and $\mathbf{a}⋅\mathbf{b}$ is dot product of vectors.

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Manifolds.log_lieMethod
log_lie(M::HeisenbergGroup, p)

Lie group logarithm for the HeisenbergGroup M of the point p. The formula reads

$$$\log\left(\begin{bmatrix} 1 & \mathbf{a} & c \\ \mathbf{0} & I_n & \mathbf{b} \\ 0 & \mathbf{0} & 1 \end{bmatrix}\right) = \begin{bmatrix} 0 & \mathbf{a} & c - \mathbf{a}⋅\mathbf{b}/2 \\ \mathbf{0} & 0_n & \mathbf{b} \\ 0 & \mathbf{0} & 0 \end{bmatrix}$$$

where $I_n$ is the $n×n$ identity matrix, $0_n$ is the $n×n$ zero matrix and $\mathbf{a}⋅\mathbf{b}$ is dot product of vectors.

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ManifoldsBase.get_coordinatesMethod
get_coordinates(M::HeisenbergGroup, p, X, ::DefaultOrthonormalBasis{ℝ,TangentSpaceType})

Get coordinates of tangent vector X at point p from the HeisenbergGroup M. Given a matrix

$$$\begin{bmatrix} 1 & \mathbf{a} & c \\ \mathbf{0} & I_n & \mathbf{b} \\ 0 & \mathbf{0} & 1 \end{bmatrix}$$$

the coordinates are concatenated vectors $\mathbf{a}$, $\mathbf{b}$, and number $c$.

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ManifoldsBase.get_vectorMethod
get_vector(M::HeisenbergGroup, p, Xⁱ, ::DefaultOrthonormalBasis{ℝ,TangentSpaceType})

Get tangent vector with coordinates Xⁱ at point p from the HeisenbergGroup M. Given a vector of coordinates $\begin{bmatrix}\mathbb{a} & \mathbb{b} & c\end{bmatrix}$ the tangent vector is equal to

$$$\begin{bmatrix} 1 & \mathbf{a} & c \\ \mathbf{0} & I_n & \mathbf{b} \\ 0 & \mathbf{0} & 1 \end{bmatrix}$$$
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ManifoldsBase.projectMethod
project(M::HeisenbergGroup{n}, p, X)

Project a matrix X in the Euclidean embedding onto the Lie algebra of HeisenbergGroup M. Sets the diagonal elements to 0 and all non-diagonal elements except the first row and the last column to 0.

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### (Special) Orthogonal and (Special) Unitary group

Since the orthogonal, unitary and special orthogonal and special unitary groups share many common functions, these are also implemented on a common level.

#### Common functions

Manifolds.exp_lieMethod
 exp_lie(G::Orthogonal{2}, X)
exp_lie(G::SpecialOrthogonal{2}, X)

Compute the Lie group exponential map on the Orthogonal(2) or SpecialOrthogonal(2) group. Given $X = \begin{pmatrix} 0 & -θ \\ θ & 0 \end{pmatrix}$, the group exponential is

$$$\exp_e \colon X ↦ \begin{pmatrix} \cos θ & -\sin θ \\ \sin θ & \cos θ \end{pmatrix}.$$$
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#### Special unitary group

Manifolds.SpecialUnitaryType
SpecialUnitary{n} = GeneralUnitaryMultiplicationGroup{n,ℝ,GeneralUnitaryMatrices{n,ℂ,DeterminantOneMatrices}}

The special unitary group $\mathrm{SU}(n)$ represented by unitary matrices of determinant +1.

The tangent spaces are of the form

$$$T_p\mathrm{SU}(x) = \bigl\{ X \in \mathbb C^{n×n} \big| X = pY \text{ where } Y = -Y^{\mathrm{H}} \bigr\}$$$

and we represent tangent vectors by just storing the SkewHermitianMatrices $Y$, or in other words we represent the tangent spaces employing the Lie algebra $\mathfrak{su}(n)$.

Constructor

SpecialUnitary(n)

Generate the Lie group of $n×n$ unitary matrices with determinant +1.

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ManifoldsBase.projectMethod
project(G::SpecialUnitary, p)

Project p to the nearest point on the SpecialUnitary group G.

Given the singular value decomposition $p = U S V^\mathrm{H}$, with the singular values sorted in descending order, the projection is

$$$\operatorname{proj}_{\mathrm{SU}(n)}(p) = U\operatorname{diag}\left[1,1,…,\det(U V^\mathrm{H})\right] V^\mathrm{H}.$$$

The diagonal matrix ensures that the determinant of the result is $+1$.

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#### Unitary group

Manifolds.UnitaryType
 Unitary{n,𝔽} = GeneralUnitaryMultiplicationGroup{n,𝔽,AbsoluteDeterminantOneMatrices}

The group of unitary matrices $\mathrm{U}(n, 𝔽)$, either complex (when 𝔽=ℂ) or quaternionic (when 𝔽=ℍ)

The group consists of all points $p ∈ 𝔽^{n × n}$ where $p^\mathrm{H}p = pp^\mathrm{H} = I$.

The tangent spaces are if the form

$$$T_p\mathrm{U}(n) = \bigl\{ X \in 𝔽^{n×n} \big| X = pY \text{ where } Y = -Y^{\mathrm{H}} \bigr\}$$$

and we represent tangent vectors by just storing the SkewHermitianMatrices $Y$, or in other words we represent the tangent spaces employing the Lie algebra $\mathfrak{u}(n, 𝔽)$.

Quaternionic unitary group is isomorphic to the compact symplectic group of the same dimension.

Constructor

Unitary(n, 𝔽::AbstractNumbers=ℂ)

Construct $\mathrm{U}(n, 𝔽)$. See also Orthogonal(n) for the real-valued case.

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Manifolds.exp_lieMethod
exp_lie(G::Unitary{2,ℂ}, X)

Compute the group exponential map on the Unitary(2) group, which is

$$$\exp_e \colon X ↦ e^{\operatorname{tr}(X) / 2} \left(\cos θ I + \frac{\sin θ}{θ} \left(X - \frac{\operatorname{tr}(X)}{2} I\right)\right),$$$

where $θ = \frac{1}{2} \sqrt{4\det(X) - \operatorname{tr}(X)^2}$.

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### Power group

Manifolds.PowerGroupMethod
PowerGroup{𝔽,T} <: GroupManifold{𝔽,<:AbstractPowerManifold{𝔽,M,RPT},ProductOperation}

Decorate a power manifold with a ProductOperation.

Constituent manifold of the power manifold must also have a IsGroupManifold or a decorated instance of one. This type is mostly useful for equipping the direct product of group manifolds with an Identity element.

Constructor

PowerGroup(manifold::AbstractPowerManifold)
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### Product group

Manifolds.ProductGroupMethod
ProductGroup{𝔽,T} <: GroupManifold{𝔽,ProductManifold{T},ProductOperation}

Decorate a product manifold with a ProductOperation.

Each submanifold must also have a IsGroupManifold or a decorated instance of one. This type is mostly useful for equipping the direct product of group manifolds with an Identity element.

Constructor

ProductGroup(manifold::ProductManifold)
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### Semidirect product group

Manifolds.SemidirectProductGroupMethod
SemidirectProductGroup(N::GroupManifold, H::GroupManifold, A::AbstractGroupAction)

A group that is the semidirect product of a normal group $\mathcal{N}$ and a subgroup $\mathcal{H}$, written $\mathcal{G} = \mathcal{N} ⋊_θ \mathcal{H}$, where $θ: \mathcal{H} × \mathcal{N} → \mathcal{N}$ is an automorphism action of $\mathcal{H}$ on $\mathcal{N}$. The group $\mathcal{G}$ has the composition rule

$$$g \circ g' = (n, h) \circ (n', h') = (n \circ θ_h(n'), h \circ h')$$$

and the inverse

$$$g^{-1} = (n, h)^{-1} = (θ_{h^{-1}}(n^{-1}), h^{-1}).$$$
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Manifolds.SemidirectProductOperationType
SemidirectProductOperation(action::AbstractGroupAction)

Group operation of a semidirect product group. The operation consists of the operation opN on a normal subgroup N, the operation opH on a subgroup H, and an automorphism action of elements of H on N. Only the action is stored.

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Manifolds.translate_diffMethod
translate_diff(G::SemidirectProductGroup, p, q, X, conX::LeftAction)

Perform differential of the left translation on the semidirect product group G.

Since the left translation is defined as (cf. SemidirectProductGroup):

$$$L_{(n', h')} (n, h) = ( L_{n'} θ_{h'}(n), L_{h'} h)$$$

then its differential can be computed as

$$$\mathrm{d}L_{(n', h')}(X_n, X_h) = ( \mathrm{d}L_{n'} (\mathrm{d}θ_{h'}(X_n)), \mathrm{d}L_{h'} X_h).$$$
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### Special Euclidean group

Manifolds.SpecialEuclideanType
SpecialEuclidean(n)

Special Euclidean group $\mathrm{SE}(n)$, the group of rigid motions.

$\mathrm{SE}(n)$ is the semidirect product of the TranslationGroup on $ℝ^n$ and SpecialOrthogonal(n)

$$$\mathrm{SE}(n) ≐ \mathrm{T}(n) ⋊_θ \mathrm{SO}(n),$$$

where $θ$ is the canonical action of $\mathrm{SO}(n)$ on $\mathrm{T}(n)$ by vector rotation.

This constructor is equivalent to calling

Tn = TranslationGroup(n)
SOn = SpecialOrthogonal(n)
SemidirectProductGroup(Tn, SOn, RotationAction(Tn, SOn))

Points on $\mathrm{SE}(n)$ may be represented as points on the underlying product manifold $\mathrm{T}(n) × \mathrm{SO}(n)$. For group-specific functions, they may also be represented as affine matrices with size (n + 1, n + 1) (see affine_matrix), for which the group operation is MultiplicationOperation.

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Manifolds.SpecialEuclideanInGeneralLinearType
SpecialEuclideanInGeneralLinear

An explicit isometric and homomorphic embedding of $\mathrm{SE}(n)$ in $\mathrm{GL}(n+1)$ and $𝔰𝔢(n)$ in $𝔤𝔩(n+1)$. Note that this is not a transparently isometric embedding.

Constructor

SpecialEuclideanInGeneralLinear(n)
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Manifolds.adjoint_actionMethod
adjoint_action(::SpecialEuclidean{3}, p, fX::TFVector{<:Any,VeeOrthogonalBasis{ℝ}})

Adjoint action of the SpecialEuclidean group on the vector with coefficients fX tangent at point p.

The formula for the coefficients reads $t×(R⋅ω) + R⋅r$ for the translation part and $R⋅ω$ for the rotation part, where t is the translation part of p, R is the rotation matrix part of p, r is the translation part of fX and ω is the rotation part of fX, $×$ is the cross product and $⋅$ is the matrix product.

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Manifolds.affine_matrixMethod
affine_matrix(G::SpecialEuclidean, p) -> AbstractMatrix

Represent the point $p ∈ \mathrm{SE}(n)$ as an affine matrix. For $p = (t, R) ∈ \mathrm{SE}(n)$, where $t ∈ \mathrm{T}(n), R ∈ \mathrm{SO}(n)$, the affine representation is the $n + 1 × n + 1$ matrix

$$$\begin{pmatrix} R & t \\ 0^\mathrm{T} & 1 \end{pmatrix}.$$$

This function embeds $\mathrm{SE}(n)$ in the general linear group $\mathrm{GL}(n+1)$. It is an isometric embedding and group homomorphism [RicoMartinez1988].

See also screw_matrix for matrix representations of the Lie algebra.

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Manifolds.exp_lieMethod
exp_lie(G::SpecialEuclidean{2}, X)

Compute the group exponential of $X = (b, Ω) ∈ 𝔰𝔢(2)$, where $b ∈ 𝔱(2)$ and $Ω ∈ 𝔰𝔬(2)$:

$$$\exp X = (t, R) = (U(θ) b, \exp Ω),$$$

where $t ∈ \mathrm{T}(2)$, $R = \exp Ω$ is the group exponential on $\mathrm{SO}(2)$,

$$$U(θ) = \frac{\sin θ}{θ} I_2 + \frac{1 - \cos θ}{θ^2} Ω,$$$

and $θ = \frac{1}{\sqrt{2}} \lVert Ω \rVert_e$ (see norm) is the angle of the rotation.

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Manifolds.exp_lieMethod
exp_lie(G::SpecialEuclidean{3}, X)

Compute the group exponential of $X = (b, Ω) ∈ 𝔰𝔢(3)$, where $b ∈ 𝔱(3)$ and $Ω ∈ 𝔰𝔬(3)$:

$$$\exp X = (t, R) = (U(θ) b, \exp Ω),$$$

where $t ∈ \mathrm{T}(3)$, $R = \exp Ω$ is the group exponential on $\mathrm{SO}(3)$,

$$$U(θ) = I_3 + \frac{1 - \cos θ}{θ^2} Ω + \frac{θ - \sin θ}{θ^3} Ω^2,$$$

and $θ = \frac{1}{\sqrt{2}} \lVert Ω \rVert_e$ (see norm) is the angle of the rotation.

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Manifolds.exp_lieMethod
exp_lie(G::SpecialEuclidean{n}, X)

Compute the group exponential of $X = (b, Ω) ∈ 𝔰𝔢(n)$, where $b ∈ 𝔱(n)$ and $Ω ∈ 𝔰𝔬(n)$:

$$$\exp X = (t, R),$$$

where $t ∈ \mathrm{T}(n)$ and $R = \exp Ω$ is the group exponential on $\mathrm{SO}(n)$.

In the screw_matrix representation, the group exponential is the matrix exponential (see exp_lie).

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Manifolds.lie_bracketMethod
lie_bracket(G::SpecialEuclidean, X::ProductRepr, Y::ProductRepr)
lie_bracket(G::SpecialEuclidean, X::AbstractMatrix, Y::AbstractMatrix)

Calculate the Lie bracket between elements X and Y of the special Euclidean Lie algebra. For the matrix representation (which can be obtained using screw_matrix) the formula is $[X, Y] = XY-YX$, while in the ProductRepr representation the formula reads $[X, Y] = [(t_1, R_1), (t_2, R_2)] = (R_1 t_2 - R_2 t_1, R_1 R_2 - R_2 R_1)$.

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Manifolds.log_lieMethod
log_lie(G::SpecialEuclidean{2}, p)

Compute the group logarithm of $p = (t, R) ∈ \mathrm{SE}(2)$, where $t ∈ \mathrm{T}(2)$ and $R ∈ \mathrm{SO}(2)$:

$$$\log p = (b, Ω) = (U(θ)^{-1} t, \log R),$$$

where $b ∈ 𝔱(2)$, $Ω = \log R ∈ 𝔰𝔬(2)$ is the group logarithm on $\mathrm{SO}(2)$,

$$$U(θ) = \frac{\sin θ}{θ} I_2 + \frac{1 - \cos θ}{θ^2} Ω,$$$

and $θ = \frac{1}{\sqrt{2}} \lVert Ω \rVert_e$ (see norm) is the angle of the rotation.

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Manifolds.log_lieMethod
log_lie(G::SpecialEuclidean{3}, p)

Compute the group logarithm of $p = (t, R) ∈ \mathrm{SE}(3)$, where $t ∈ \mathrm{T}(3)$ and $R ∈ \mathrm{SO}(3)$:

$$$\log p = (b, Ω) = (U(θ)^{-1} t, \log R),$$$

where $b ∈ 𝔱(3)$, $Ω = \log R ∈ 𝔰𝔬(3)$ is the group logarithm on $\mathrm{SO}(3)$,

$$$U(θ) = I_3 + \frac{1 - \cos θ}{θ^2} Ω + \frac{θ - \sin θ}{θ^3} Ω^2,$$$

and $θ = \frac{1}{\sqrt{2}} \lVert Ω \rVert_e$ (see norm) is the angle of the rotation.

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Manifolds.log_lieMethod
log_lie(G::SpecialEuclidean{n}, p) where {n}

Compute the group logarithm of $p = (t, R) ∈ \mathrm{SE}(n)$, where $t ∈ \mathrm{T}(n)$ and $R ∈ \mathrm{SO}(n)$:

$$$\log p = (b, Ω),$$$

where $b ∈ 𝔱(n)$ and $Ω = \log R ∈ 𝔰𝔬(n)$ is the group logarithm on $\mathrm{SO}(n)$.

In the affine_matrix representation, the group logarithm is the matrix logarithm (see log_lie):

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Manifolds.screw_matrixMethod
screw_matrix(G::SpecialEuclidean, X) -> AbstractMatrix

Represent the Lie algebra element $X ∈ 𝔰𝔢(n) = T_e \mathrm{SE}(n)$ as a screw matrix. For $X = (b, Ω) ∈ 𝔰𝔢(n)$, where $Ω ∈ 𝔰𝔬(n) = T_e \mathrm{SO}(n)$, the screw representation is the $n + 1 × n + 1$ matrix

$$$\begin{pmatrix} Ω & b \\ 0^\mathrm{T} & 0 \end{pmatrix}.$$$

This function embeds $𝔰𝔢(n)$ in the general linear Lie algebra $𝔤𝔩(n+1)$ but it's not a homomorphic embedding (see SpecialEuclideanInGeneralLinear for a homomorphic one).

See also affine_matrix for matrix representations of the Lie group.

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Manifolds.translate_diffMethod
translate_diff(G::SpecialEuclidean, p, q, X, ::RightAction)

Differential of the right action of the SpecialEuclidean group on itself. The formula for the rotation part is the differential of the right rotation action, while the formula for the translation part reads

$$$R_q⋅X_R⋅t_p + X_t$$$

where $R_q$ is the rotation part of q, $X_R$ is the rotation part of X, $t_p$ is the translation part of p and $X_t$ is the translation part of X.

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### Special linear group

Manifolds.SpecialLinearType
SpecialLinear{n,𝔽} <: AbstractDecoratorManifold

The special linear group $\mathrm{SL}(n,𝔽)$ that is, the group of all invertible matrices with unit determinant in $𝔽^{n×n}$.

The Lie algebra $𝔰𝔩(n, 𝔽) = T_e \mathrm{SL}(n,𝔽)$ is the set of all matrices in $𝔽^{n×n}$ with trace of zero. By default, tangent vectors $X_p ∈ T_p \mathrm{SL}(n,𝔽)$ for $p ∈ \mathrm{SL}(n,𝔽)$ are represented with their corresponding Lie algebra vector $X_e = p^{-1}X_p ∈ 𝔰𝔩(n, 𝔽)$.

The default metric is the same left-$\mathrm{GL}(n)$-right-$\mathrm{O}(n)$-invariant metric used for GeneralLinear(n, 𝔽). The resulting geodesic on $\mathrm{GL}(n,𝔽)$ emanating from an element of $\mathrm{SL}(n,𝔽)$ in the direction of an element of $𝔰𝔩(n, 𝔽)$ is a closed subgroup of $\mathrm{SL}(n,𝔽)$. As a result, most metric functions forward to GeneralLinear.

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ManifoldsBase.projectMethod
project(G::SpecialLinear, p, X)

Orthogonally project $X ∈ 𝔽^{n × n}$ onto the tangent space of $p$ to the SpecialLinear $G = \mathrm{SL}(n, 𝔽)$. The formula reads

$$$\operatorname{proj}_{p} = (\mathrm{d}L_p)_e ∘ \operatorname{proj}_{𝔰𝔩(n, 𝔽)} ∘ (\mathrm{d}L_p^{-1})_p \colon X ↦ X - \frac{\operatorname{tr}(X)}{n} I,$$$

where the last expression uses the tangent space representation as the Lie algebra.

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ManifoldsBase.projectMethod
project(G::SpecialLinear, p)

Project $p ∈ \mathrm{GL}(n, 𝔽)$ to the SpecialLinear group $G=\mathrm{SL}(n, 𝔽)$.

Given the singular value decomposition of $p$, written $p = U S V^\mathrm{H}$, the formula for the projection is

$$$\operatorname{proj}_{\mathrm{SL}(n, 𝔽)}(p) = U S D V^\mathrm{H},$$$

where

$$$D_{ij} = δ_{ij} \begin{cases} 1 & \text{ if } i ≠ n \\ \det(p)^{-1} & \text{ if } i = n \end{cases}.$$$
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### Translation group

Manifolds.TranslationGroupType
TranslationGroup{T<:Tuple,𝔽} <: GroupManifold{Euclidean{T,𝔽},AdditionOperation}

Translation group $\mathrm{T}(n)$ represented by translation arrays.

Constructor

TranslationGroup(n₁,...,nᵢ; field = 𝔽)

Generate the translation group on $𝔽^{n₁,…,nᵢ}$ = Euclidean(n₁,...,nᵢ; field = 𝔽), which is isomorphic to the group itself.

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## Group actions

Group actions represent actions of a given group on a specified manifold. The following operations are available:

Furthermore, group operation action features the following:

The following group actions are available:

Manifolds.adjoint_apply_diff_groupMethod
adjoint_apply_diff_group(A::AbstractGroupAction, a, X, p)

Pullback with respect to group element of group action A.

$$$(\mathrm{d}τ^{p,*}) : T_{τ_{a} p} \mathcal M → T_{a} \mathcal G$$$
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Manifolds.apply!Method
apply!(A::AbstractGroupAction, q, a, p)

Apply action a to the point p with the rule specified by A. The result is saved in q.

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Manifolds.applyMethod
apply(A::AbstractGroupAction, a, p)

Apply action a to the point p using map $τ_a$, specified by A. Unless otherwise specified, the right action is defined in terms of the left action:

$$$\mathrm{R}_a = \mathrm{L}_{a^{-1}}$$$
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Manifolds.apply_diffMethod
apply_diff(A::AbstractGroupAction, a, p, X)

For group point $p ∈ \mathcal M$ and tangent vector $X ∈ T_p \mathcal M$, compute the action on $X$ of the differential of the action of $a ∈ \mathcal{G}$, specified by rule A. Written as $(\mathrm{d}τ_a)_p$, with the specified left or right convention, the differential transports vectors

$$$(\mathrm{d}τ_a)_p : T_p \mathcal M → T_{τ_a p} \mathcal M$$$
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Manifolds.apply_diff_groupMethod
apply_diff_group(A::AbstractGroupAction, a, X, p)

Compute the value of differential of action AbstractGroupAction A on vector X, where element a is acting on p, with respect to the group element.

Let $\mathcal G$ be the group acting on manifold $\mathcal M$ by the action A. The action is of element $g ∈ \mathcal G$ on a point $p ∈ \mathcal M$. The differential transforms vector X from the tangent space at a ∈ \mathcal G, $X ∈ T_a \mathcal G$ into a tangent space of the manifold $\mathcal M$. When action on element p is written as $\mathrm{d}τ^p$, with the specified left or right convention, the differential transforms vectors

$$$(\mathrm{d}τ^p) : T_{a} \mathcal G → T_{τ_a p} \mathcal M$$$

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Manifolds.center_of_orbitFunction
center_of_orbit(
A::AbstractGroupAction,
pts,
p,
)

Calculate an action element $a$ of action A that is the mean element of the orbit of p with respect to given set of points pts. The mean is calculated using the method mean_method.

The orbit of $p$ with respect to the action of a group $\mathcal{G}$ is the set

$$$O = \{ τ_a p : a ∈ \mathcal{G} \}.$$$

This function is useful for computing means on quotients of manifolds by a Lie group action.

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Manifolds.inverse_apply!Method
inverse_apply!(A::AbstractGroupAction, q, a, p)

Apply inverse of action a to the point p with the rule specified by A. The result is saved in q.

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Manifolds.inverse_applyMethod
inverse_apply(A::AbstractGroupAction, a, p)

Apply inverse of action a to the point p. The action is specified by A.

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Manifolds.inverse_apply_diffMethod
inverse_apply_diff(A::AbstractGroupAction, a, p, X)

For group point $p ∈ \mathcal M$ and tangent vector $X ∈ T_p \mathcal M$, compute the action on $X$ of the differential of the inverse action of $a ∈ \mathcal{G}$, specified by rule A. Written as $(\mathrm{d}τ_a^{-1})_p$, with the specified left or right convention, the differential transports vectors

$$$(\mathrm{d}τ_a^{-1})_p : T_p \mathcal M → T_{τ_a^{-1} p} \mathcal M$$$
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Manifolds.optimal_alignment!Method
optimal_alignment!(A::AbstractGroupAction, x, p, q)

Calculate an action element of action A that acts upon p to produce the element closest to q. The result is written to x.

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Manifolds.optimal_alignmentMethod
optimal_alignment(A::AbstractGroupAction, p, q)

Calculate an action element $a$ of action A that acts upon p to produce the element closest to q in the metric of the G-manifold:

$$$\arg\min_{a ∈ \mathcal{G}} d_{\mathcal M}(τ_a p, q)$$$

where $\mathcal{G}$ is the group that acts on the G-manifold $\mathcal M$.

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### Rotation action

Manifolds.ColumnwiseMultiplicationActionType
ColumnwiseMultiplicationAction{
TM<:AbstractManifold,
TO<:GeneralUnitaryMultiplicationGroup,
} <: AbstractGroupAction{TAD}

Action of the (special) unitary or orthogonal group GeneralUnitaryMultiplicationGroup of type On columns of points on a matrix manifold M.

Constructor

ColumnwiseMultiplicationAction(
M::AbstractManifold,
On::GeneralUnitaryMultiplicationGroup,
)
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Manifolds.RowwiseMultiplicationActionType
RowwiseMultiplicationAction{
TM<:AbstractManifold,
TO<:GeneralUnitaryMultiplicationGroup,
} <: AbstractGroupAction{TAD}

Action of the (special) unitary or orthogonal group GeneralUnitaryMultiplicationGroup of type On columns of points on a matrix manifold M.

Constructor

RowwiseMultiplicationAction(
M::AbstractManifold,
On::GeneralUnitaryMultiplicationGroup,
)
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Manifolds.applyMethod
apply(A::RotationAroundAxisAction, θ, p)

Rotate point p from Euclidean(3) manifold around axis A.axis by angle θ. The formula reads

$$$p_{rot} = (\cos(θ))p + (k×p) \sin(θ) + k (k⋅p) (1-\cos(θ)),$$$

where $k$ is the vector A.axis and ⋅ is the dot product.

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Manifolds.optimal_alignmentMethod
optimal_alignment(A::LeftColumnwiseMultiplicationAction, p, q)

Compute optimal alignment for the left ColumnwiseMultiplicationAction, i.e. the group element $O^{*}$ that, when it acts on p, returns the point closest to q. Details of computation are described in Section 2.2.1 of [Srivastava2016].

$$$O^{*} = \begin{cases} UV^T & \text{if } \operatorname{det}(p q^{\mathrm{T}})\\ U K V^{\mathrm{T}} & \text{otherwise} \end{cases}$$$

where $U \Sigma V^{\mathrm{T}}$ is the SVD decomposition of $p q^{\mathrm{T}}$ and $K$ is the unit diagonal matrix with the last element on the diagonal replaced with -1.

References

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## Metrics on groups

Lie groups by default typically forward all metric-related operations like exponential or logarithmic map to the underlying manifold, for example SpecialOrthogonal uses methods for Rotations (which is, incidentally, bi-invariant), or SpecialEuclidean uses product metric of the translation and rotation parts (which is not invariant under group operation).

It is, however, possible to change the metric used by a group by wrapping it in a MetricManifold decorator.

### Invariant metrics

Manifolds.directionMethod
direction(::AbstractDecoratorManifold) -> AD

Get the direction of the action a certain Lie group with its implicit metric has

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Manifolds.has_approx_invariant_metricMethod
has_approx_invariant_metric(
G::AbstractDecoratorManifold,
p,
X,
Y,
qs::AbstractVector,
conv::ActionDirection = LeftAction();
kwargs...,
) -> Bool

Check whether the metric on the group $\mathcal{G}$ is (approximately) invariant using a set of predefined points. Namely, for $p ∈ \mathcal{G}$, $X,Y ∈ T_p \mathcal{G}$, a metric $g$, and a translation map $τ_q$ in the specified direction, check for each $q ∈ \mathcal{G}$ that the following condition holds:

$$$g_p(X, Y) ≈ g_{τ_q p}((\mathrm{d}τ_q)_p X, (\mathrm{d}τ_q)_p Y).$$$

This is necessary but not sufficient for invariance.

Optionally, kwargs passed to isapprox may be provided.

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## Cartan-Schouten connections

Manifolds.CartanSchoutenMinusType
CartanSchoutenMinus

The unique Cartan-Schouten connection such that all left-invariant vector fields are globally defined by their value at identity. It is biinvariant with respect to the group operation.

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Manifolds.CartanSchoutenPlusType
CartanSchoutenPlus

The unique Cartan-Schouten connection such that all right-invariant vector fields are globally defined by their value at identity. It is biinvariant with respect to the group operation.

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Manifolds.CartanSchoutenZeroType
CartanSchoutenZero

The unique torsion-free Cartan-Schouten connection. It is biinvariant with respect to the group operation.

If the metric on the underlying manifold is bi-invariant then it is equivalent to the Levi-Civita connection of that metric.

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Base.expMethod
exp(M::ConnectionManifold{𝔽,<:AbstractDecoratorManifold{𝔽},<:AbstractCartanSchoutenConnection}, p, X) where {𝔽}

Compute the exponential map on the ConnectionManifold M with a Cartan-Schouten connection. See Sections 5.3.2 and 5.3.3 of [Pennec2020] for details.

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Base.logMethod
log(M::ConnectionManifold{𝔽,<:AbstractDecoratorManifold{𝔽},<:AbstractCartanSchoutenConnection}, p, q) where {𝔽}

Compute the logarithmic map on the ConnectionManifold M with a Cartan-Schouten connection. See Sections 5.3.2 and 5.3.3 of [Pennec2020] for details.

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• Suhubi2013

E. Suhubi, Exterior Analysis: Using Applications of Differential Forms, 1st edition. Amsterdam: Academic Press, 2013.

• AndruchowLarotondaRechtVarela2014

Andruchow E., Larotonda G., Recht L., and Varela A.: “The left invariant metric in the general linear group”, Journal of Geometry and Physics 86, pp. 241-257, 2014. doi: 10.1016/j.geomphys.2014.08.009, arXiv: 1109.0520v1.

• MartinNeff2016

Martin, R. J. and Neff, P.: “Minimal geodesics on GL(n) for left-invariant, right-O(n)-invariant Riemannian metrics”, Journal of Geometric Mechanics 8(3), pp. 323-357, 2016. doi: 10.3934/jgm.2016010, arXiv: 1409.7849v2.

• BinzPods2008

E. Binz and S. Pods, The Geometry of Heisenberg Groups: With Applications in Signal Theory, Optics, Quantization, and Field Quantization. American Mathematical Soc., 2008.

• Gallier2002

Gallier J.; Xu D.; Computing exponentials of skew-symmetric matrices and logarithms of orthogonal matrices. International Journal of Robotics and Automation (2002), 17(4), pp. 1-11. pdf.

• Andrica2013

Andrica D.; Rohan R.-A.; Computing the Rodrigues coefficients of the exponential map of the Lie groups of matrices. Balkan Journal of Geometry and Its Applications (2013), 18(2), pp. 1-2. pdf.

• RicoMartinez1988

Rico Martinez, J. M., “Representations of the Euclidean group and its applications to the kinematics of spatial chains,” PhD Thesis, University of Florida, 1988.

• Srivastava2016

A. Srivastava and E. P. Klassen, Functional and Shape Data Analysis. Springer New York, 2016. ISBN: 978-1-4939-4018-9. doi: 10.1007/978-1-4939-4020-2.

• Pennec2020

X. Pennec and M. Lorenzi, “5 - Beyond Riemannian geometry: The affine connection setting for transformation groups,” in Riemannian Geometric Statistics in Medical Image Analysis, X. Pennec, S. Sommer, and T. Fletcher, Eds. Academic Press, 2020, pp. 169–229. doi: 10.1016/B978-0-12-814725-2.00012-1.