Vector bundles

CotangentSpaceAtPoint(M::Manifold, p)

Return an object of type VectorSpaceAtPoint representing cotangent space at p.

FVector(type::VectorSpaceType, data)

Decorator indicating that the vector data is from a fiber of a vector bundle of type type.

TangentSpaceAtPoint(M::Manifold, p)

Return an object of type VectorSpaceAtPoint representing tangent space at p on the Manifold M.

TensorProductType(spaces::VectorSpaceType...)

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

VectorBundle{𝔽,TVS<:VectorSpaceType,TM<:Manifold{𝔽}} <: Manifold{𝔽}

Vector bundle on a Manifold M of type VectorSpaceType.

Constructor

VectorBundle(M::Manifold, type::VectorSpaceType)

VectorBundleFibers(fiber::VectorSpaceType, M::Manifold)

Type representing a family of vector spaces (fibers) of a vector bundle over M with vector spaces of type fiber. In contrast with VectorBundle, operations on VectorBundleFibers expect point-like and vector-like parts to be passed separately instead of being bundled together. It can be thought of as a representation of vector spaces from a vector bundle but without storing the point at which a vector space is attached (which is specified separately in various functions).

VectorBundleVectorTransport(
    method_point::AbstractVectorTransportMethod,
    method_vector::AbstractVectorTransportMethod,
)

Vector transport type on VectorBundle. method_point is used for vector transport of the point part and method_vector is used for transport of the vector part

VectorSpaceAtPoint{
    𝔽,
    TFiber<:VectorBundleFibers{<:VectorSpaceType,<:Manifold{𝔽}},
    TX,
} <: Manifold{𝔽}

A vector space at a point p on the manifold. This is modelled using VectorBundleFibers with only a vector-like part and fixing the point-like part to be just p.

This vector space itself is also a manifold. Especially, it's flat and hence isometric to the Euclidean manifold.

Constructor

VectorSpaceAtPoint(fiber::VectorBundleFibers, p)

A vector space (fiber type fiber of a vector bundle) at point p from the manifold fiber.manifold.

VectorSpaceType

Abstract type for tangent spaces, cotangent spaces, their tensor products, exterior products, etc.

Every vector space fiber is supposed to provide:

• a method of constructing vectors,
• basic operations: addition, subtraction, multiplication by a scalar and negation (unary minus),
• zero_vector!(fiber, X, p) to construct zero vectors at point p,
• allocate(X) and allocate(X, T) for vector X and type T,
• copyto!(X, Y) for vectors X and Y,
• number_eltype(v) for vector v,
• vector_space_dimension(::VectorBundleFibers{<:typeof(fiber)}) where fiber.

Optionally:

• inner product via inner (used to provide Riemannian metric on vector bundles),
• flat and sharp,
• norm (by default uses inner),
• project (for embedded vector spaces),
• representation_size (if support for ProductArray is desired),
• broadcasting for basic operations.

exp(B::VectorBundle, p, X)

Exponential map of tangent vector $X$ at point $p$ from 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 $X = (V_{X,M}, V_{X,F}) ∈ T_pB$ 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 $F$).

The exponential map is calculated as

\[\exp_p(X) = (\exp_{x_p}(V_{X,M}), V_{\exp})\]

where $V_{\exp}$ is the result of vector transport of $V_p + V_{X,F}$ to the point $\exp_{x_p}(V_{X,M})$. The sum $V_p + V_{X,F}$ corresponds to the exponential map in the vector space $F$.

exp(M::TangentSpaceAtPoint, p, X)

Exponential map of tangent vectors X and p from the tangent space M. It is calculated as their sum.

getindex(p::ProductRepr, M::VectorBundle, s::Symbol)
p[M::VectorBundle, s]

Access the element(s) at index s of a point p on a VectorBundle M by using the symbols :point and :vector for the base and vector component, respectively.

log(B::VectorBundle, p, q)

Logarithmic map of the point y at point p from 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 $B$. Similarly, $q = (x_q, V_q)$.

The logarithmic map is calculated as

\[\log_p q = (\log_{x_p}(x_q), V_{\log} - V_p)\]

where $V_{\log}$ is the result of vector transport of $V_q$ to the point $x_p$. The difference $V_{\log} - V_p$ corresponds to the logarithmic map in the vector space $F$.

log(M::TangentSpaceAtPoint, p, q)

Logarithmic map on the tangent space manifold M, calculated as the difference of tangent vectors q and p from M.

setindex!(p::ProductRepr, val, M::VectorBundle, s::Symbol)
p[M::VectorBundle, s] = val

Set the element(s) at index s of a point p on a VectorBundle M to val by using the symbols :point and :vector for the base and vector component, respectively.

norm(B::VectorBundleFibers, p, q)

Norm of the vector X from the vector space of type B.fiber at point p from manifold B.manifold.

bundle_projection(B::VectorBundle, x::ProductRepr)

Projection of point p from the bundle M to the base manifold. Returns the point on the base manifold B.manifold at which the vector part of p is attached.

flat(M::Manifold, p, X::FVector)

Compute the flat isomorphism (one of the musical isomorphisms) of tangent vector X from the vector space of type M at point p from the underlying Manifold.

The function can be used for example to transform vectors from the tangent bundle to vectors from the cotangent bundle $♭ : T\mathcal M → T^{*}\mathcal M$

sharp(M::Manifold, p, ξ::FVector)

Compute the sharp isomorphism (one of the musical isomorphisms) of vector ξ from the vector space M at point p from the underlying Manifold.

The function can be used for example to transform vectors from the cotangent bundle to vectors from the tangent bundle $♯ : T^{*}\mathcal M → T\mathcal M$

vector_bundle_transport(fiber::VectorSpaceType, M::Manifold)

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

vector_space_dimension(B::VectorBundleFibers)

Dimension of the vector space of type B.

zero_vector!(B::VectorBundleFibers, X, p)

Save the zero vector from the vector space of type B.fiber at point p from manifold B.manifold to X.

zero_vector(B::VectorBundleFibers, p)

Compute the zero vector from the vector space of type B.fiber at point p from manifold B.manifold.

allocate_result(B::VectorBundleFibers, f, x...)

Allocates an array for the result of function f that is an element of the vector space of type B.fiber on manifold B.manifold and arguments x... for implementing the non-modifying operation using the modifying operation.

allocate_result_type(B::VectorBundleFibers, f, args::NTuple{N,Any}) where N

Returns type of element of the array that will represent the result of function f for representing an operation with result in the vector space fiber for manifold M on given arguments (passed at a tuple).

distance(B::VectorBundle, p, q)

Distance between points $x$ and $y$ 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 $B$. Similarly, $q = (x_q, V_q)$.

The distance is calculated as

\[d_B(x, y) = \sqrt{d_M(x_p, x_q)^2 + d_F(V_p, V_{q←p})^2}\]

where $d_\mathcal M$ is the distance on manifold $\mathcal M$, $d_F$ is the distance between two vectors from the fiber $F$ and $V_{q←p}$ is the result of parallel transport of vector $V_q$ to point $x_p$. The default behavior of vector_transport_to is used to compute the vector transport.

distance(B::VectorBundleFibers, p, X, Y)

Distance between vectors X and Y from the vector space at point p from the manifold B.manifold, that is the base manifold of M.

distance(M::TangentSpaceAtPoint, p, q)

Distance between vectors p and q from the vector space M. It is calculated as the norm of their difference.

injectivity_radius(M::TangentSpaceAtPoint)

Return the injectivity radius on the TangentSpaceAtPoint M, which is $∞$.

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

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

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

Inner product of vectors X and Y from the vector space of type B.fiber at point p from manifold B.manifold.

inner(M::TangentSpaceAtPoint, p, X, Y)

Inner product of vectors X and Y from the tangent space at M.

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 $T_{x_p}\mathcal M$.

project(B::VectorBundleFibers, p, X)

Project vector X from the vector space of type B.fiber at point p.

project(M::TangentSpaceAtPoint, p, X)

Project the vector X from the tangent space M, that is project the vector X tangent at M.point.

project(M::TangentSpaceAtPoint, p)

Project the point p from the tangent space M, that is project the vector p tangent at M.point.

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

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

zero_tangent_vector(B::VectorBundle, p)

Zero tangent vector at point p from the vector bundle B over manifold B.fiber (denoted $\mathcal M$). The zero vector belongs to the 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 zero vector is calculated as

\[\mathbf{0}_{p} = (\mathbf{0}_{x_p}, \mathbf{0}_F)\]

where $\mathbf{0}_{x_p}$ is the zero tangent vector from $T_{x_p}\mathcal M$ and $\mathbf{0}_F$ is the zero element of the vector space $F$.

zero_tangent_vector(M::TangentSpaceAtPoint, p)

Zero tangent vector at point p from the tangent space M, that is the zero tangent vector at point M.point.