# 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.inFunction
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.

source
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.

source

# Fallback for the exponential map: Solving the corresponding ODE

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

ManifoldsBase.inverse_retractMethod
inverse_retract(M, p, q method::NLSolveInverseRetraction; kwargs...)

Approximate the inverse of the retraction specified by method.retraction from p with respect to q on the AbstractManifold M using NLsolve. This inverse retraction is not guaranteed to succeed and probably will not unless q is close to p and the initial guess X0 is close.

If the solver fails to converge, an OutOfInjectivityRadiusError is raised. See NLSolveInverseRetraction for configurable parameters.

source

# Public documentation

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

Manifolds.submanifold_componentFunction
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.

source
Manifolds.submanifold_componentsFunction
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.

source
Manifolds.ProductReprType
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.

source

## 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.