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 ArrayPartition (from RecursiveArrayTools.jl) with canonical projections $Π_i : \mathcal{M} → \mathcal{M}_i$ for $i ∈ 1, 2, …, n$ provided by submanifold_component.
ManifoldDiff.riemannian_Hessian — MethodY = riemannian_Hessian(M::ProductManifold, p, G, H, X)
riemannian_Hessian!(M::ProductManifold, Y, p, G, H, X)Compute the Riemannian Hessian $\operatorname{Hess} f(p)[X]$ given the Euclidean gradient $∇ f(\tilde p)$ in G and the Euclidean Hessian $∇^2 f(\tilde p)[\tilde X]$ in H, where $\tilde p, \tilde X$ are the representations of $p,X$ in the embedding,.
On a product manifold, this decouples and can be computed elementwise.
Manifolds.flat — Methodflat(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.
Manifolds.manifold_volume — Methodmanifold_volume(M::ProductManifold)Return the volume of ProductManifold M, i.e. product of volumes of the manifolds M is constructed from.
Manifolds.sharp — Methodsharp(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
Manifolds.volume_density — Methodvolume_density(M::ProductManifold, p, X)Return volume density on the ProductManifold M, i.e. product of constituent volume densities.