Combinations of Manifolds

# Special Manifolds build upon one or more Riemannian manifolds

## Tangent bundle

The tangent bundle $T\mathcal M$ of a manifold $\mathcal M$ consists of all tuples $(x,\xi) \in T\mathcal M$, where $\xi\in T_x\mathcal M$, $x\in \mathcal M$, where the metric is inherited component wise and for the exponential and logarithmic map, the second component requires a parallelTransport.

### Tangent Bundle Types

TBPoint <: MPoint

A point $N\in \mathcal M$ on the manifold $\mathcal M = T\mathcal N$ represented by a tuple (x,ξ), where $x\in\mathcal N$ is a point on the manifold and $\xi=\xi_x\in T_x\mathcal N$ is a point in the tangent space at $x$.

Two constructors are available:

• TBPoint(x,ξ) to construct a tangent bundle point by specifying both an MPoint x and a TVector ξ.
• TBPoint( (X) ) to construct a tangent bundle point from a tuple X=(x,ξ),

i.e. the value of another tangent bundle point.

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TBTVector <: TVector

A tangent vector $\Xi \in T_X\mathcal M$ on the manifold $\mathcal M = T\mathcal N$ for the (base) manifold $\mathcal N$. Both tangent components can be represented by elements from the base point $x$ from within $X=(x,\xi)$. Both components are from the same space since $TT_x\mathcal N= T_x\mathcal N$, hence the tangent vector is a tuple $(\xi\,\nu)\in T_x\mathcal N\times T_x\mathcal N$. As for the TBPoint two constructors are available, one for stwo seperate tangent vectors, one for a tuple of two tangent vectors.

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TangentBundle <: Manifold

The manifold $\mathcal M = T\mathcal N$ obtained by looking at the tangent bundle of a Manifolds tangent spaces. The manifold obtained is of dimension $2d$, where $d$ is the dimension of the manifold $\mathcal N$ considered.

To keep notations clear, small letters will always refer to points (x,y) or tangent vectors (ξ,η) on the manifold $\mathcal N$, while capital letters (X, Y, Z and Ξ,Η) will refer to points and tangent vectors in the tangent bundle respectively.

Abbreviation

TB

Constructor

TangentBundle(M)

generates the tangent bundle to the Manifold M.

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### Tangent Bundle Functions

exp(M,X,Ξ[, t=1.0])

Compute the exponential map on the TangentBundle M$=T\mathcal N$ with respect to the TBPointX=(x,ξ) and the TBTVectorΞ=(Ξx,Ξξ), which consists of the exponential map in the first component (exp(x,Ξx,t) and a (scaled) addition in the second (ξ + tΞξ) in the second component followed by a parallel transport.

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log(M,X,Y)

Compute the logarithmic map on the TangentBundle $\mathcal M=T\mathcal N$, i.e. the log for the base manifold component and a parallel transport and a minus for the tangent components.

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dot(M,X,Ξ,Η)

Compute the Riemannian inner product for two TBTVectors Ξ and Η from $T_X\mathcal M$ of the TangentBundleM = TN given by the sum of the two inner products of the tangent vector components

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norm(M,X,Ξ)

Computes the norm of the TBTVectorΞ in the tangent space $T_x\mathcal M$ at TBPointX of the TangentBundle M.

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distance(M,X,Y)

Compute the Riemannian distance on $\mathcal M=T\mathcal N$ by employing the distance on the manifold for the base component and the vector norm on the tangent space, and then take the Eucklidean Norm of the vector from $\mathbb R^2$.

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getBase(Ξ)

return the base of the TBTVectorΞ, i.e. its first TVector.

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getBase(M)

return the base manifold of the TangentBundle Manifold M.

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getBase(X)

return the base of the TBPointX, i.e. its MPoint.

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getTangent(Ξ)

return the tangent of the TBTVectorΞ, i.e. its second TBTVector.

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getTangent(X)

return the tangent of the TBPointX, i.e. the its TVector.

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getValue(Ξ)

return the Tuple contained in the TBTVectorΞ, i.e. its tuple of two TVectors.

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getValue(X)

return the value of the TBPoint X, i.e. the Tuple of a MPoint and its TVector.

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manifoldDimension(X)

returns the dimension of the TangentBundle M$=T\mathcal N$ to which X bvelongs, which is twice the dimension of the base manifold.

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manifoldDimension(M)

returns the dimension of the TangentBundle M$=T\mathcal N$, i.e., twice the dimension of the base manifold N.

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parallelTransport(M,X,Y,Ξ)

Compute the paralllel transport of the TBTVectorΞ from the tangent space $T_X\mathcal M$ at TBPointX to $T_Y\mathcal M$ at TBPointY on the TangentBundle M provided that the corresponding geodesic $g(\cdot;x,y)$ is unique. Then both components of $\Xi=(\Xi_x,\Xi_\xi)$ are parallely transported using the parallel transport of the underlying base manifold.

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randomMPoint(M)

returns a random point on the TangentBundle M by producing a randomMPoint random point on the base manifold and randomTVector in the correspoinding tangent plane.

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randomTVector(M,x)

returns a random tangent vector the TangentBundle M by producing two randomTVectors in the correspoinding tangent plane of the getBase of the TBPoint x.

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tangentONB(M,X,Y)

constructs a tangent ONB in the tangent space of the TBPointX on the TangentBundle M, where $\log_XY$ is the first component.

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Η,κ = tangentONB(M,X,Ξ)

constructs a tangent ONB in $T_X\mathcal M$, i.e. in the tangent space of the TBPoint x on the TangentBundle M whose first vector is given by the TBTVectorΞ. It is constructed by using twice the tangent ONB of the base manifold.

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typicalDistance(M)

returns the typical distance on the TangentBundle M, i.e. for $\mathcal M = T\mathcal N$ we obtain $t_{\mathcal M} = \sqrt{t_{\mathcal N}^2 + d_{\mathcal N}^2}$, where $d$ denotes the manifold dimension.

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validateMPoint(M,X)

validate that the TBPointX is a valid point on the TangentBundle M, i.e. the first component is a point on the base manifold and the second a tangent vector is the tangent space of the first

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validateTVector(M,X,Ξ)

validate, that the TBTVectorΞ is a valid tangent vector in the tangent space of the TBPointX on the TangentBundle M, i.e. both components of Ξ are tangent vectors in the tangent space of the base component of X, since the tangent space of the tangent space is represented as the tangent space itself.

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zeroTVector(M,X)

returns a zero vector in the tangent space $T_X\mathcal M$ of the TangentBundle $X=(x,ξ)\in T\mathcal N$ by creating two zero vectors in $T_x\mathcal M$.

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## Power Manifold

The product manifold $\mathcal M^n$, where $n\in\mathbb N^k$ represents arrays that are manifold-valued, for example, if $n$ is a number ($k=1$) we obtain a manifold-valued signal $f\in\mathcal M^n$. Many operations are performed element wise, while for example the distance on the power manifold is the $\ell^2$ norm of the element wise distances.

### Power Manifold Types

PowPoint <: MPoint

A point on the power manifold $\mathcal M = \mathcal N^n$ represented by an array (of size n) of MPoints.

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PowTVector <: TVector

A tangent vector on the power manifold $\mathcal M = \mathcal N^n$ represented by an array (of size n) of TVectors.

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Power{M<:Manifold} <: Manifold

A power manifold $\mathcal M = \mathcal N^n$, where $n$ can be an integer or an integer vector.

Abbreviation

Pow

Constructors

Power(M,n)

construct the power manifold $\mathcal M^n$ for a Manifold M and a natural number n.

Power(M,n)

construct the power manifold $\mathcal M^{n_1\times n_2\times\cdots\times n_d}$ for a Manifold M and a Tuple or Array n of natural numbers.

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### Power Manifold Functions

exp(M,x,ξ[, t=1.0])

compute the product exponential map on the Power manifold M and return the corresponding PowPoint.

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log(M,x,y)

compute the product logarithmic map on the Power manifold M and return the corresponding PowTVector.

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dot(M,x,ξ,ν)

compute the inner product as sum of the component inner products on the Power manifold M.

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norm(M,x,ξ)

compute the norm of the [PowTVector] ξ induced by the metric on the manifold components of the Power manifold M.

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distance(M,x,y)

compute a vectorized version of distance on the [Power] manifold M for two PowPoint x and y.

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manifoldDimension(x)

return the (product of) dimension(s) of the Power the PowPointx belongs to.

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manifoldDimension(M)

return the (product of) dimension(s) of the Power manifold M.

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parallelTransport(M,x,y,ξ)

compute the product parallelTransport map on the Power manifold M from the PowPoint x to y of the PowTVector ξ.

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randomMPoint(M)

construct a random point on the Power manifold M, by creating n points on the Manifold M.manifold as corresponding PowPoint. Optional values are passed down.

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randomTVector(M,x)

construct a random tangent vector on the Power manifold M, by creating n tangent vectors on the Manifold M.manifold at the enrties of the PowPoint x. Optional values are passed down.

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(Ξ,κ) = tangentONB(M,x,y)

compute an ONB within the tangent space $T_x\mathcal M$ such that $\xi=\log_xy$ is the first vector and compute the eigenvalues of the curvature tensor $R(\Xi,\dot g)\dot g$, where $g=g_{x,\xi}$ is the geodesic with $g(0)=x$, $\dot g(0) = \xi$, i.e. $\kappa_1$ corresponding to $\Xi_1=\xi$ is zero.

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typicalDistance(M)

returns the typical distance on the Power manifold M, which is based on the elementwise manifold.

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ξ = zeroTVector(M,x)

returns a zero vector in the tangent space $T_x\mathcal M$ of the PowPoint $x\in\mathcal M$ on the Power manifold M.

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## Product Manifold

A little more general is the product manifold, where $\mathcal M = \mathcal N_1\times\cdots\times\mathcal N_n$, $n\in\mathbb N^k$ is a product of manifolds, i.e. for a value $f\in\mathcal M$ we have that $f_i\in\mathcal N_i$, where $i$ might be a multi-index.

### Product Manifold Types

ProdPoint <: MPoint

A point on the Product $\mathcal M = \mathcal N_1\times\mathcal N_2\times\cdots\times\mathcal N_m$,$m\in\mathbb N$, represented by a vector or array of MPoints.

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ProdTVector <: TVector

A tangent vector in the product of tangent spaces of the Product $T\mathcal M = T\mathcal N_1\times T\mathcal N_2\times\cdots\times T\mathcal N_m$,$m\in\mathbb N$, represented by a vector or array of TVectors.

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Product{M<:Manifold} <: Manifold

a product manifold $\mathcal M = \mathcal N_1\times\mathcal N_2\times\cdots\times\mathcal N_m$, $m\in\mathbb N$, concatinates a set of manifolds $\mathcal N_i$, $i=1,\ldots,m$, into one using the sum of the metrics to impose a metric on this manifold. The manifold can also be an arbitrary Array of manifolds, not necessarily only a vector.

Abbreviation

Prod

Constructor

Product(m)

constructs a Power Manifold based on an array m of Manifolds.

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### Product Manifold Functions

exp(M,x,ξ)

computes the product exponential map on the Product manifold M and returns the corresponding ProdPoint.

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log(M,x,y)

computes the product logarithmic map from PowPoint x to y on the Product manifold M and returns the corresponding ProdTVector.

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dot(M,x,ξ,ν)

compute the inner product as sum of the component inner products on the Product manifold M.

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norm(M,x,ξ)

norm of the ProdTVector ξ induced by the metric on the manifold components of the Product manifold M.

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distance(M,x,y)

compute a vectorized version of distance for two ProdPoints x and y on the Product manifold M.

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manifoldDimension(x)

returns the (product of) dimension(s) of the Product manifold M the ProdPoint x belongs to.

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manifoldDimension(M)

returns the (product of) dimension(s) of the Product manifold M.

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parallelTransport(M,x,ξ)

computes the product parallelTransport map on the Product manifold M and returns the corresponding ProdTVector.

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randomMPoint(M)

generate a random point on Product manifold M.

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randomTVector(M,x)

generate a random tangent vector in the tangent space of the ProdPoint x on Power manifold M.

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typicalDistance(M)

returns the typical distance on Product manifold M, which is the minimum of the internal ones.

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validateMPoint(M,x)

validate, that the ProdPoint x is a point on the Product manifold M, i.e. that the array dimensions are correct and that all elements are valid points on each elements manifolds

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validateTVector(M,x,ξ)

validate, that the ProdTVector ξ is a valid tangent vector to the ProdPoint x on the Product manifold M, i.e. that all three array dimensions match and this validation holds elementwise.

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ξ = zeroTVector(M,x)

returns a zero vector in the tangent space $T_x\mathcal M$ of the ProdPoint $x\in\mathcal M$ on the Product manifold M.

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## Graph Manifold

The Graph manifold provides methods for two often interacting manifolds on a given graph $\mathcal G = (\mathcal V,\mathcal E)$: A vertex graph manifold, $\mathcal M^{\lvert \mathcal V\rvert}$ and an edge manifold $\mathcal N^{\lvert \mathcal E\rvert}$ for two Manifolds $\mathcal M$ and $\mathcal N$. For example $\mathcal N$ might be the tangent bundle of $\mathcal M$.

### Types

Graph{M<:Manifold} <: Manifold

The graph manifold models manifold-valued data on a graph $\mathcal G = (\mathcal V, \mathcal E)$, both on vertices and edges as well as their interplay. The adjacency is stored in a matrix, and may contain also the weights.

Since there are two possibilities in dimensions, $\lvert\mathcal V\rvert$ and $\lvert\mathcal E\rvert$, the manifold itself will refer to the first one, while depending on the type of MPoint one of them is returned.

Fields

the default values are given in brackets

• adjacency – the (sparse) adjacency matrix, might also carry weights, i.e. all $a_{ij}>0$ refer to adjacent nodes $i$ and $j$
• name – (A Graph manifold of \$Submanifold.) name of the manifold • manifold – the internal manifold present at vertices (edges) for • dimension – stores the dimension of the manifold of a GraphVertexPoint • isDirected – (false) indicates whether the graph is directed or not. source GraphEdgePoint <: MPoint A point graph edge power manifold$\mathcal M = \mathcal N^{\lvert\mathcal E\rvert}$represented by a (sparse/not completely filled matrix of corresponding MPoints. source GraphEdgeTVector <: TVector A tangent vector$\xi\in T_x\mathcal M$to the graph edge power manifold$\mathcal M = \mathcal N^{\lvert\mathcal E\rvert}$represented by a (sparse/not completely filled) matrix of corresponding TVectors. source GraphVertexPoint <: MPoint A point graph vertex power manifold$\mathcal M = \mathcal N^{\lvert\mathcal V\rvert}$represented by a vector of corresponding MPoints. source GraphVertexTVector <: TVector A tangent vector$\xi\in T_x\mathcal M$to the graph vertex power manifold$\mathcal M = \mathcal N^{\lvert\mathcal V\rvert}$represented by a vector of corresponding TVectors. source ### Functions exp(M,x,ξ[, t=1.0]) computes the product exponential map on the Graph vertices and returns the corresponding GraphVertexPoint. source log(M,x,y) computes the product logarithmic map on the Graph for two GraphEdgePoint x and y and returns the corresponding GraphEdgeTVector. source log(M,x,y) computes the product logarithmic map on the Graph for two GraphVertexPoint x and y and returns the corresponding GraphVertexTVector. source dot(M,x,ξ,ν) computes the inner product as sum of the component inner products on the Graph edges. source dot(M,x,ξ,ν) computes the inner product as sum of the component inner products on the Graph vertices. source norm(M,x,ξ) norm of the GraphEdgeTVector ξ induced by the metric on the manifold components of the Graph manifold M. source norm(M,x,ξ) norm of the GraphVertexTVector ξ induced by the metric on the manifold components of the Graph manifold M. source distance(M,x,y) compute a vectorized version of distance on the Graph manifold M for two GraphEdgePoint x and y. source distance(M,x,y) compute a vectorized version of distance on the Graph manifold M for two GraphVertexPoint x and y. source manifoldDimension(x) returns the (product of) dimension(s) of the Graph manifold M the GraphEdgePointx belongs to. source manifoldDimension(x) returns the (product of) dimension(s) of the Graph manifold M the GraphVertexPoint x belongs to. source manifoldDimension(M) returns the (product of) dimension(s) of the Graph manifold M seen as a vertex power manifold. source parallelTransport(M,x,ξ) compute the product parallelTransport map on the Graph edge power manifold and returns the corresponding GraphVertexTVector. source parallelTransport(M,x,ξ) compute the product parallelTransport map on the Graph vertex power manifold$\mathcal M^{\lvert\mathcal V\rvert}$and returns the corresponding GraphVertexTVector. source randomMPoint(M,[,:Vertex]) compute a random point on the Graph manifold, where by default a point on the vertices is produces, use :Edge to generate a GraphEdgePoint. Further optional parameters are passed on to the element wise random point function. source randomTVector(M,x) compute a random GraphEdgeTVector to the GraphVertexPoint x on the Graph manifold M by calling the inner random vector generation for every edge source ξ = zeroTVector(M,x) returns a zero vector in the tangent space$T_x\mathcal M$of the GraphEdgePoint$x\in\mathcal M$on the Graph edge manifold M. source ξ = zeroTVector(M,x) returns a zero vector in the tangent space$T_x\mathcal M$of the GraphVertexPoint$x\in\mathcal M\$ on the Graph vertex manifold M.

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startEdgePoint(M,x)

For a Graph manifold and a GraphVertexPoint, this function constructs the corresponding GraphEdgePoint, such that each edge has its start point vertex value assigned.

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sumEdgeTVector(M,ξ)

return the GraphVertexTVector where edge tangents are summed in their start point.

For an GraphEdgeTVector ξ on a Graph manifold M this function assumes that all edge tangents are attached in a tangent space corresponding to the same point on the base manifold, i.e. all these vectors can be summed. This sum per vectex is then returned as a GraphVertexTVector.

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