Quotient manifold

IsQuotientManifold <: AbstractTrait

Specify that a certain decorated manifold is a quotient manifold in the sense that it provides implicitly (or explicitly through QuotientManifold properties of a quotient manifold.

See QuotientManifold for more details.

QuotientManifold{M <: AbstractManifold{𝔽}, N} <: AbstractManifold{𝔽}

Equip a manifold $\mathcal M$ explicitly with the property of being a quotient manifold.

A manifold $\mathcal M$ is then a a quotient manifold of another manifold $\mathcal N$, i.e. for an equivalence relation $∼$ on $\mathcal N$ we have

\[ \mathcal M = \mathcal N / ∼ = \bigl\{ [p] : p ∈ \mathcal N \bigr\},\]

where $[p] ≔ \{ q ∈ \mathcal N : q ∼ p\}$ denotes the equivalence class containing $p$. For more details see Subsection 3.4.1 [AMS08].

This manifold type models an explicit quotient structure. This should be done if either the default implementation of $\mathcal M$ uses another representation different from the quotient structure or if it provides a (default) quotient structure that is different from the one introduced here.

Fields

• manifold – the manifold $\mathcal M$ in the introduction above.
• total_space – the manifold $\mathcal N$ in the introduction above.

Constructor

QuotientManifold(M,N)

Create a manifold where M is the quotient manifold and Nis its total space.

canonical_project!(M, q, p)

Compute the canonical projection $π$ on a manifold $\mathcal M$ that IsQuotientManifold, e.g. a QuotientManifold in place of q.

See canonical_project for more details.

canonical_project(M, p)

Compute the canonical projection $π$ on a manifold $\mathcal M$ that IsQuotientManifold, e.g. a QuotientManifold. The canonical (or natural) projection $π$ from the total space $\mathcal N$ onto $\mathcal M$ given by

\[ π = π_{\mathcal N, \mathcal M} : \mathcal N → \mathcal M, p ↦ π_{\mathcal N, \mathcal M}(p) = [p].\]

in other words, this function implicitly assumes, that the total space $\mathcal N$ is given, for example explicitly when M is a QuotientManifold and p is a point on N.

differential_canonical_project!(M, Y, p, X)

Compute the differential of the canonical projection $π$ on a manifold $\mathcal M$ that IsQuotientManifold, e.g. a QuotientManifold. See differential_canonical_project for details.

differential_canonical_project(M, p, X)

Compute the differential of the canonical projection $π$ on a manifold $\mathcal M$ that IsQuotientManifold, e.g. a QuotientManifold. The canonical (or natural) projection $π$ from the total space $\mathcal N$ onto $\mathcal M$, such that its differential

\[ Dπ(p) : T_p\mathcal N → T_{π(p)}\mathcal M\]

where again the total space might be implicitly assumed, or explicitly when using a QuotientManifold M. So here p is a point on N and X is from $T_p\mathcal N$.

get_orbit_action(M::AbstractDecoratorManifold)

Return the group action that generates the orbit of an equivalence class of the quotient manifold M for which equivalence classes are orbits of an action of a Lie group. For the case that

\[\mathcal M = \mathcal N / \mathcal O,\]

where $\mathcal O$ is a Lie group with its group action generating the orbit.

get_total_space(M::AbstractDecoratorManifold)

Return the total space of a manifold that IsQuotientManifold, e.g. a QuotientManifold.

horizontal_component(N::AbstractManifold, p, X)
horizontal_compontent(QuotientManifold{𝔽,M,N}, p, X)

Compute the horizontal component of tangent vector X at point p in the total space of quotient manifold N.

This is often written as the space $\mathrm{Hor}_p^π\mathcal N$.

horizontal_lift!(N, Y, q, X)
horizontal_lift!(QuotientManifold{𝔽,M,N}, Y, p, X)

Compute the horizontal lift of X from $T_p\mathcal M$, $p=π(q)$. to `T_q\mathcal N in place of Y.

horizontal_lift(N::AbstractManifold, q, X)
horizontal_lift(::QuotientManifold{𝔽,M,N}, p, X)

Given a point q in total space of quotient manifold N such that $p=π(q)$ is a point on a quotient manifold M (implicitly given for the first case) and a tangent vector X this method computes a tangent vector Y on the horizontal space of $T_q\mathcal N$, i.e. the subspace that is orthogonal to the kernel of $Dπ(q)$.

vertical_component(N::AbstractManifold, p, X)
vertical_component(QuotientManifold{𝔽,M,N}, p, X)

Compute the vertical component of tangent vector X at point p in the total space of quotient manifold N.

This is often written as the space $\mathrm{ver}_p^π\mathcal N$.