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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.ProductManifold — Type
ProductManifold{𝔽,TM<:Tuple} <: Manifold{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.exp — Method
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.log — Method
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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Manifolds.flat — Method
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.sharp — Method
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.submanifold — Method
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_tangent_vector — Method
check_tangent_vector(M::ProductManifold, p, X; check_base_point = true, kwargs... )

Check whether X is a tangent vector to p on the ProductManifold M, i.e. after check_manifold_point(M, p), and all projections to base manifolds must be respective tangent vectors.

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

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ManifoldsBase.distance — Method
distance(M::ProductManifold, p, q)

Compute the distance between two points p and q on the ProductManifold M, which is the 2-norm of the elementwise distances on the internal manifolds that build M.

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ManifoldsBase.inner — Method
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_retract — Method
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.retract — Method
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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Manifolds.ProductFVectorDistribution — Type
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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LinearAlgebra.cross — Method
cross(M,N)
cross(M1, M2, M3,...)

Return the ProductManifold For two Manifolds 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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