# Product manifold

Product manifold $\mathcal M = \mathcal{M}_1 × \mathcal{M}_2 × … × \mathcal{M}_n$ of manifolds $\mathcal{M}_1, \mathcal{M}_2, …, \mathcal{M}_n$. Points on the product manifold can be constructed using ProductRepr with canonical projections $Π_i : \mathcal{M} → \mathcal{M}_i$ for $i ∈ 1, 2, …, n$ provided by submanifold_component.

Manifolds.ProductFVectorDistributionType
ProductFVectorDistribution([type::VectorBundleFibers], [x], distrs...)

Generates a random vector at point x from vector space (a fiber of a tangent bundle) of type type using the product distribution of given distributions.

Vector space type and x can be automatically inferred from distributions distrs.

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Manifolds.ProductManifoldType
ProductManifold{𝔽,TM<:Tuple} <: AbstractManifold{𝔽}

Product manifold $M_1 × M_2 × … × M_n$ with product geometry.

Constructor

ProductManifold(M_1, M_2, ..., M_n)

generates the product manifold $M_1 × M_2 × … × M_n$. Alternatively, the same manifold can be contructed using the × operator: M_1 × M_2 × M_3.

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

compute the exponential map from p in the direction of X on the ProductManifold M, which is the elementwise exponential map on the internal manifolds that build M.

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Base.logMethod
log(M::ProductManifold, p, q)

Compute the logarithmic map from p to q on the ProductManifold M, which can be computed using the logarithmic maps of the manifolds elementwise.

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Base.randMethod
rand(M::ProductManifold; parts_kwargs = map(_ -> (;), M.manifolds))

Return a random point on ProductManifold M. parts_kwargs is a tuple of keyword arguments for rand on each manifold in M.manifolds.

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LinearAlgebra.crossMethod
cross(M, N)
cross(M1, M2, M3,...)

Return the ProductManifold For two AbstractManifolds M and N, where for the case that one of them is a ProductManifold itself, the other is either prepended (if N is a product) or appenden (if M) is. If both are product manifold, they are combined into one product manifold, keeping the order.

For the case that more than one is a product manifold of these is build with the same approach as above

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Manifolds.change_metricMethod
change_metric(M::ProductManifold, ::AbstractMetric, p, X)

Since the metric on a product manifold decouples, the change of metric can be done elementwise.

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Manifolds.change_representerMethod
change_representer(M::ProductManifold, ::AbstractMetric, p, X)

Since the metric on a product manifold decouples, the change of a representer can be done elementwise

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Manifolds.flatMethod
flat(M::ProductManifold, p, X::FVector{TangentSpaceType})

use the musical isomorphism to transform the tangent vector X from the tangent space at p on the ProductManifold M to a cotangent vector. This can be done elementwise for every entry of X (with respect to the corresponding entry in p) separately.

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Manifolds.sharpMethod
sharp(M::ProductManifold, p, ξ::FVector{CotangentSpaceType})

Use the musical isomorphism to transform the cotangent vector ξ from the tangent space at p on the ProductManifold M to a tangent vector. This can be done elementwise for every entry of ξ (and p) separately

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Manifolds.submanifoldMethod
submanifold(M::ProductManifold, i::Val)
submanifold(M::ProductManifold, i::AbstractVector)

Extract the factor of the product manifold M indicated by indices in i. For example, for i equal to Val((1, 3)) the product manifold constructed from the first and the third factor is returned.

The version with AbstractVector is not type-stable, for better preformance use Val.

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ManifoldsBase.check_vectorMethod
check_vector(M::ProductManifold, p, X; kwargs... )

Check whether X is a tangent vector to p on the ProductManifold M, i.e. all projections to base manifolds must be respective tangent vectors. If X is not a tangent vector to p on M a CompositeManifoldError.consisting of all error messages of the components, for which the tests fail is returned.

The tolerance for the last test can be set using the kwargs....

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ManifoldsBase.innerMethod
inner(M::ProductManifold, p, X, Y)

compute the inner product of two tangent vectors X, Y from the tangent space at p on the ProductManifold M, which is just the sum of the internal manifolds that build M.

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ManifoldsBase.inverse_retractMethod
inverse_retract(M::ProductManifold, p, q, m::InverseProductRetraction)

Compute the inverse retraction from p with respect to q on the ProductManifold M using an InverseProductRetraction, which by default encapsulates a inverse retraction for each manifold of the product. Then this method is performed elementwise, so the encapsulated inverse retraction methods have to be available per factor.

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ManifoldsBase.retractMethod
retract(M::ProductManifold, p, X, m::ProductRetraction)

Compute the retraction from p with tangent vector X on the ProductManifold M using an ProductRetraction, which by default encapsulates retractions of the base manifolds. Then this method is performed elementwise, so the encapsulated retractions method has to be one that is available on the manifolds.

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