# Vector bundles

Vector bundle $E$ is a manifold that is built on top of another manifold $\mathcal M$ (base space). It is characterized by a continuous function $Π : E → \mathcal M$, such that for each point $p ∈ \mathcal M$ the preimage of $p$ by $Π$, $Π^{-1}(\{p\})$, has a structure of a vector space. These vector spaces are called fibers. Bundle projection can be performed using function bundle_projection.

Tangent bundle is a simple example of a vector bundle, where each fiber is the tangent space at the specified point $x$. An object representing a tangent bundle can be obtained using the constructor called TangentBundle.

Fibers of a vector bundle are represented by the type VectorBundleFibers. The important difference between functions operating on VectorBundle and VectorBundleFibers is that in the first case both a point on the underlying manifold and the vector are represented together (by a single argument) while in the second case only the vector part is present, while the point is supplied in a different argument where needed.

VectorBundleFibers refers to the whole set of fibers of a vector bundle. There is also another type, VectorSpaceAtPoint, that represents a specific fiber at a given point. This distinction is made to reduce the need to repeatedly construct objects of type VectorSpaceAtPoint in certain usage scenarios. This is also considered a manifold.

## FVector

For cases where confusion between different types of vectors is possible, the type FVector can be used to express which type of vector space the vector belongs to. It is used for example in musical isomorphisms (the flat and sharp functions) that are used to go from a tangent space to cotangent space and vice versa.

## Documentation

Manifolds.SasakiRetractionType
struct SasakiRetraction <: AbstractRetractionMethod end

Exponential map on TangentBundle computed via Euler integration as described in [Muralidharan2012]. The system of equations for $\gamma : ℝ \to T\mathcal M$ such that $\gamma(1) = \exp_{p,X}(X_M, X_F)$ and $\gamma(0)=(p, X)$ reads

$$$\dot{\gamma}(t) = (\dot{p}(t), \dot{X}(t)) = (R(X(t), \dot{X}(t))\dot{p}(t), 0)$$$

where $R$ is the Riemann curvature tensor (see riemann_tensor).

Constructor

SasakiRetraction(L::Int)

In this constructor L is the number of integration steps.

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Manifolds.TangentBundleType
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 [Sasaki1958].

Exact retraction and inverse retraction can be approximated using VectorBundleProductRetraction, VectorBundleInverseProductRetraction and SasakiRetraction. VectorBundleProductVectorTransport can be used as a vector transport.

Constructors

TangentBundle(M::AbstractManifold)
TangentBundle(M::AbstractManifold, vtm::VectorBundleProductVectorTransport)
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Manifolds.VectorBundleFibersType
VectorBundleFibers(fiber::VectorSpaceType, M::AbstractManifold)

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).

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Manifolds.VectorBundleInverseProductRetractionType
struct VectorBundleInverseProductRetraction <: AbstractInverseRetractionMethod end

Inverse retraction of the point y at point p from vector bundle B over manifold B.fiber (denoted $\mathcal M$). The inverse retraction is derived as a product manifold-style approximation to the logarithmic map in the Sasaki metric. The considered product manifold is the product between the manifold $\mathcal M$ and the topological vector space isometric to the fiber.

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 inverse retraction is calculated as

$$$\operatorname{retr}^{-1}_p q = (\operatorname{retr}^{-1}_{x_p}(x_q), V_{\operatorname{retr}^{-1}} - V_p)$$$

where $V_{\operatorname{retr}^{-1}}$ is the result of vector transport of $V_q$ to the point $x_p$. The difference $V_{\operatorname{retr}^{-1}} - V_p$ corresponds to the logarithmic map in the vector space $F$.

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Manifolds.VectorBundleProductRetractionType
struct VectorBundleProductRetraction <: AbstractRetractionMethod end

Product retraction map of tangent vector $X$ at point $p$ from vector bundle B over manifold B.fiber (denoted $\mathcal M$). The retraction is derived as a product manifold-style approximation to the exponential map in the Sasaki metric. The considered product manifold is the product between the manifold $\mathcal M$ and the topological vector space isometric to the fiber.

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 retraction is calculated as

$$$\operatorname{retr}_p(X) = (\exp_{x_p}(V_{X,M}), V_{\exp})$$$

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 $F$.

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Manifolds.VectorBundleProductVectorTransportType
VectorBundleProductVectorTransport{
TMP<:AbstractVectorTransportMethod,
TMV<:AbstractVectorTransportMethod,
} <: 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.

The vector transport is derived as a product manifold-style vector transport. The considered product manifold is the product between the manifold $\mathcal M$ and the topological vector space isometric to the fiber.

Constructor

VectorBundleProductVectorTransport(
method_point::AbstractVectorTransportMethod,
method_vector::AbstractVectorTransportMethod,
)
VectorBundleProductVectorTransport()

By default both methods are set to ParallelTransport.

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Manifolds.VectorSpaceAtPointType
VectorSpaceAtPoint{
𝔽,
TFiber<:VectorBundleFibers{<:VectorSpaceType,<:AbstractManifold{𝔽}},
TX,
} <: AbstractManifold{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.

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Base.expMethod
exp(M::TangentSpaceAtPoint, p, X)

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

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Base.getindexMethod
getindex(p::ArrayPartition, 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.

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Base.getindexMethod
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.

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Base.logMethod
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.

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Base.setindex!Method
setindex!(p::ArrayPartition, 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.

Note

The content of element of p is replaced, not the element itself.

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Base.setindex!Method
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.

Note

The content of element of p is replaced, not the element itself.

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LinearAlgebra.normMethod
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.

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Manifolds.bundle_projectionMethod
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.

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ManifoldsBase.allocate_resultMethod
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.

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

Return 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).

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ManifoldsBase.distanceMethod
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.

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ManifoldsBase.distanceMethod
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.

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ManifoldsBase.innerMethod
inner(M::TangentSpaceAtPoint, p, X, Y)

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

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ManifoldsBase.innerMethod
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

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

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ManifoldsBase.projectMethod
project(M::TangentSpaceAtPoint, p, X)

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

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ManifoldsBase.projectMethod
project(M::TangentSpaceAtPoint, p)

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

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ManifoldsBase.projectMethod
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 $F$.

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ManifoldsBase.projectMethod
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 $T_{x_p}\mathcal M$.

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ManifoldsBase.projectMethod
project(B::VectorBundleFibers, p, X)

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

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ManifoldsBase.zero_vector!Method
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.

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ManifoldsBase.zero_vectorMethod
zero_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.

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ManifoldsBase.zero_vectorMethod
zero_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 $F$.

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ManifoldsBase.zero_vectorMethod
zero_vector(B::VectorBundleFibers, p)

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

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## Example

The following code defines a point on the tangent bundle of the sphere $S^2$ and a tangent vector to that point.

using Manifolds
M = Sphere(2)
TB = TangentBundle(M)
p = ProductRepr([1.0, 0.0, 0.0], [0.0, 1.0, 3.0])
X = ProductRepr([0.0, 1.0, 0.0], [0.0, 0.0, -2.0])
ProductRepr with 2 submanifold components:
Component 1 =
3-element Vector{Float64}:
0.0
1.0
0.0
Component 2 =
3-element Vector{Float64}:
0.0
0.0
-2.0

An approximation of the exponential in the Sasaki metric using 1000 steps can be calculated as follows.

q = retract(TB, p, X, SasakiRetraction(1000))
println("Approximation of the exponential map: ", q)
Approximation of the exponential map: ProductRepr{Tuple{Vector{Float64}, Vector{Float64}}}(([0.6759570857309888, 0.35241486404386485, 0.6472138609849252], [-1.0318269583261073, 0.6273324630574116, 0.7360618920075961]))
• Muralidharan2012

P. Muralidharan and P. T. Fletcher, “Sasaki Metrics for Analysis of Longitudinal Data on Manifolds,” Proc IEEE Comput Soc Conf Comput Vis Pattern Recognit, vol. 2012, pp. 1027–1034, Jun. 2012, doi: 10.1109/CVPR.2012.6247780.

• Sasaki1958

S. Sasaki, “On the differential geometry of tangent bundles of Riemannian manifolds,” Tohoku Math. J. (2), vol. 10, no. 3, pp. 338–354, 1958, doi: 10.2748/tmj/1178244668.