Vector bundles

TensorProductType(spaces::VectorSpaceType...)

Vector space type corresponding to the tensor product of given vector space types.

TangentBundle{𝔽,M} = VectorBundle{𝔽,TangentSpaceType,M} where {𝔽,M<:AbstractManifold{𝔽}}

Tangent bundle for manifold of type M, as a manifold with the Sasaki metric [Sas58].

Exact retraction and inverse retraction can be approximated using FiberBundleProductRetraction, FiberBundleInverseProductRetraction and SasakiRetraction. FiberBundleProductVectorTransport can be used as a vector transport.

Constructors

TangentBundle(M::AbstractManifold)
TangentBundle(M::AbstractManifold, vtm::FiberBundleProductVectorTransport)

VectorBundle{𝔽,TVS,TM,VTV} = FiberBundle{𝔽,TVS,TM,TVT} where {TVS<:VectorSpaceType}

Alias for FiberBundle when fiber type is a TVS of type VectorSpaceType.

VectorSpaceFiberType is used to encode vector spaces as fiber types.

fiber_bundle_transport(M::AbstractManifold, fiber::FiberType)

Determine the vector transport used for exp and log maps on a vector bundle with fiber type fiber and manifold M.

injectivity_radius(M::TangentBundle)

Injectivity radius of TangentBundle manifold is infinite if the base manifold is flat and 0 otherwise. See https://mathoverflow.net/questions/94322/injectivity-radius-of-the-sasaki-metric.

inner(B::VectorBundle, p, X, Y)

Inner product of tangent vectors X and Y at point p from the vector bundle B over manifold B.fiber (denoted $\mathcal M$).

Notation:

• The point $p = (x_p, V_p)$ where $x_p ∈ \mathcal M$ and $V_p$ belongs to the fiber $F=π^{-1}(\{x_p\})$ of the vector bundle $B$ where $π$ is the canonical projection of that vector bundle $B$.
• The tangent vector $v = (V_{X,M}, V_{X,F}) ∈ T_{x}B$ where $V_{X,M}$ is a tangent vector from the tangent space $T_{x_p}\mathcal M$ and $V_{X,F}$ is a tangent vector from the tangent space $T_{V_p}F$ (isomorphic to $F$). Similarly for the other tangent vector $w = (V_{Y,M}, V_{Y,F}) ∈ T_{x}B$.

The inner product is calculated as

$⟨X, Y⟩_p = ⟨V_{X,M}, V_{Y,M}⟩_{x_p} + ⟨V_{X,F}, V_{Y,F}⟩_{V_p}.$

inverse_retract(M::VectorBundle, p, q, ::FiberBundleInverseProductRetraction)

Compute the allocating variant of the FiberBundleInverseProductRetraction, which by default allocates and calls inverse_retract_product!.

is_flat(::VectorBundle)

Return true if the underlying manifold of VectorBundle M is flat.

project(B::VectorBundle, p, X)

Project the element X of the ambient space of the tangent space $T_p B$ to the tangent space $T_p B$.

Notation:

• The point $p = (x_p, V_p)$ where $x_p ∈ \mathcal M$ and $V_p$ belongs to the fiber $F=π^{-1}(\{x_p\})$ of the vector bundle $B$ where $π$ is the canonical projection of that vector bundle $B$.
• The vector $x = (V_{X,M}, V_{X,F})$ where $x_p$ belongs to the ambient space of $T_{x_p}\mathcal M$ and $V_{X,F}$ belongs to the ambient space of the fiber $F=π^{-1}(\{x_p\})$ of the vector bundle $B$ where $π$ is the canonical projection of that vector bundle $B$.

The projection is calculated by projecting $V_{X,M}$ to tangent space $T_{x_p}\mathcal M$ and then projecting the vector $V_{X,F}$ to the fiber $F$.

project(B::VectorBundle, p)

Project the point p from the ambient space of the vector bundle B over manifold B.fiber (denoted $\mathcal M$) to the vector bundle.

Notation:

• The point $p = (x_p, V_p)$ where $x_p$ belongs to the ambient space of $\mathcal M$ and $V_p$ belongs to the ambient space of the fiber $F=π^{-1}(\{x_p\})$ of the vector bundle $B$ where $π$ is the canonical projection of that vector bundle $B$.

The projection is calculated by projecting the point $x_p$ to the manifold $\mathcal M$ and then projecting the vector $V_p$ to the tangent space $T_{x_p}\mathcal M$.

retract(M::VectorBundle, p, q, t::Number, ::FiberBundleProductRetraction)

Compute the allocating variant of the FiberBundleProductRetraction, which by default allocates and calls retract_product!.

vector_transport_to(M::VectorBundle, p, X, q, m::FiberBundleProductVectorTransport)

Compute the vector transport the tangent vector Xat p to q on the VectorBundle M using the FiberBundleProductVectorTransport m.