Ease of notation

The following terms introduce a nicer notation for some operations, for example using the ∈ operator, $p ∈ \mathcal M$, to determine whether $p$ is a point on the AbstractManifold $\mathcal M$.

Base.in — Function

Base.in(p, M::AbstractManifold; kwargs...)
p ∈ M

Check, whether a point p is a valid point (i.e. in) a AbstractManifold M. This method employs is_point deactivating the error throwing option.

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Base.in(p, TpM::TangentSpaceAtPoint; kwargs...)
X ∈ TangentSpaceAtPoint(M,p)

Check whether X is a tangent vector from (in) the tangent space $T_p\mathcal M$, i.e. the TangentSpaceAtPoint at p on the AbstractManifold M. This method uses is_vector deactivating the error throw option.

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ManifoldsBase.TangentSpace — Constant

TangentSpace(M::AbstractManifold, p)

Return a TangentSpaceAtPoint representing tangent space at p on the AbstractManifold M.

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Fallback for the exponential map: Solving the corresponding ODE

When additionally loading NLSolve.jl the following fallback for the exponential map is available.

Public documentation

The following functions are of interest for extending and using the ProductManifold.

Manifolds.submanifold_component — Function

submanifold_component(M::AbstractManifold, p, i::Integer)
submanifold_component(M::AbstractManifold, p, ::Val(i)) where {i}
submanifold_component(p, i::Integer)
submanifold_component(p, ::Val(i)) where {i}

Project the product array p on M to its ith component. A new array is returned.

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Manifolds.submanifold_components — Function

submanifold_components(M::AbstractManifold, p)
submanifold_components(p)

Get the projected components of p on the submanifolds of M. The components are returned in a Tuple.

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Manifolds.ProductRepr — Type

ProductRepr(parts)

A more general but slower representation of points and tangent vectors on a product manifold.

Example:

A product point on a product manifold Sphere(2) × Euclidean(2) might be created as

ProductRepr([1.0, 0.0, 0.0], [2.0, 3.0])

where [1.0, 0.0, 0.0] is the part corresponding to the sphere factor and [2.0, 3.0] is the part corresponding to the euclidean manifold.

Warning

ProductRepr is deprecated and will be removed in a future release. Please use ArrayPartition instead.

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Specific exception types

For some manifolds it is useful to keep an extra index, at which point on the manifold, the error occurred as well as to collect all errors that occurred on a manifold. This page contains the manifold-specific error messages this package introduces.