# Atlases and charts

Atlases on an $n$-dimensional manifold $\mathcal M$ are collections of charts $\mathcal A = \{(U_i, φ_i) \colon i \in I\}$, where $I$ is a (finite or infinte) index family, such that $U_i \subseteq \mathcal M$ is an open set and each chart $φ_i: U_i \to \mathbb{R}^n$ is a homeomorphism. This means, that $φ_i$ is bijecive – sometimes also called one-to-one and onto - and continuous, and its inverse $φ_i^{-1}$ is continuous as well. The inverse $φ_i^{-1}$ is called (local) parametrization. The resulting parameters $a=φ(p)$ of $p$ (with respect to the chart $φ$) are in the literature also called “(local) coordinates”. To distinguish the parameter $a$ from get_coordinates in a basis, we use the terminology parameter in this package.

For an atlas $\mathcal A$ we further require that

$$$\displaystyle\bigcup_{i\in I} U_i = \mathcal M.$$$

We say that $φ_i$ is a chart about $p$, if $p\in U_i$. An atlas provides a connection between a manifold and the Euclidean space $\mathbb{R}^n$, since locally, a chart about $p$ can be used to identify its neighborhood (as long as you stay in $U_i$) with a subset of a Euclidean space. Most manifolds we consider are smooth, i.e. any change of charts $φ_i \circ φ_j^{-1}: \mathbb{R}^n\to\mathbb{R}^n$, where $i,j\in I$, is a smooth function. These changes of charts are also called transition maps.

Most operations on manifolds in Manifolds.jl avoid operating in a chart through appropriate embeddings and formulas derived for particular manifolds, though atlases provide the most general way of working with manifolds. Compared to these approaches, using an atlas is often more technical and time-consuming. They are extensively used in metric-related functions on MetricManifolds.

Atlases are represented by objects of subtypes of AbstractAtlas. There are no type restrictions for indices of charts in atlases.

Operations using atlases and charts are available through the following functions:

• get_chart_index can be used to select an appropriate chart for the neighborhood of a given point $p$. This function should work deterministically, i.e. for a fixed $p$ always return the same chart.
• get_parameters converts a point to its parameters with respect to the chart in a chart.
• get_point converts parameters (local coordinates) in a chart to the point that corresponds to them.
• induced_basis returns a basis of a given vector space at a point induced by a chart $φ$.
• transition_map converts coordinates of a point between two charts, e.g. computes $φ_i\circ φ_j^{-1}: \mathbb{R}^n\to\mathbb{R}^n$, $i,j\in I$.

While an atlas could store charts as explicit functions, it is favourable, that the [get_parameters] actually implements a chart $φ$, get_point its inverse, the prametrization $φ^{-1}$.

Manifolds.InducedBasisType
InducedBasis(vs::VectorSpaceType, A::AbstractAtlas, i)

The basis induced by chart with index i from an AbstractAtlas A of vector space of type vs.

For the vs a TangentSpace this works as follows:

Let $n$ denote the dimension of the manifold $\mathcal M$.

Let the parameter $a=φ_i(p) ∈ \mathbb R^n$ and $j∈\{1,…,n\}$. We can look at the $j$th parameter curve $b_j(t) = a + te_j$, where $e_j$ denotes the $j$th unit vector. Using the parametrisation we obtain a curve $c_j(t) = φ_i^{-1}(b_j(t))$ which fulfills $c(0) = p$.

Now taking the derivative(s) with respect to $t$ (and evaluate at $t=0$), we obtain a tangent vector for each $j$ corresponding to an equivalence class of curves (having the same derivative) as

$$$X_j = [c_j] = \frac{\mathrm{d}}{\mathrm{d}t} c_i(t) \Bigl|_{t=0}$$$

and the set $\{X_1,\ldots,X_n\}$ is the chart-induced basis of $T_p\mathcal M$.

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Manifolds.RetractionAtlasType
RetractionAtlas{
𝔽,
TRetr<:AbstractRetractionMethod,
TInvRetr<:AbstractInverseRetractionMethod,
TBasis<:AbstractBasis,
} <: AbstractAtlas{𝔽}

An atlas indexed by points on a manifold, $\mathcal M = I$ and parameters (local coordinates) are given in $T_p\mathcal M$. This means that a chart $φ_p = \mathrm{cord}\circ\mathrm{retr}_p^{-1}$ is only locally defined (around $p$), where $\mathrm{cord}$ is the decomposition of the tangent vector into coordinates with respect to the given basis of the tangent space, cf. get_coordinates. The parametrization is given by $φ_p^{-1}=\mathrm{retr}_p\circ\mathrm{vec}$, where $\mathrm{vec}$ turns the basis coordinates into a tangent vector, cf. get_vector.

In short: The coordinates with respect to a basis are used together with a retraction as a parametrization.

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Manifolds.get_parametersMethod
get_parameters(M::AbstractManifold, A::AbstractAtlas, i, p)

Calculate parameters (local coordinates) of point p on manifold M in chart from an AbstractAtlas A at index i. This function is hence an implementation of the chart $φ_i(p), i\in I$. The parameters are in the number system determined by A. If the point $p\notin U_i$ is not in the domain of the chart, this method should throw an error.

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Manifolds.get_pointMethod
get_point(M::AbstractManifold, A::AbstractAtlas, i, a)

Calculate point at parameters (local coordinates) a on manifold M in chart from an AbstractAtlas A at index i. This function is hence an implementation of the inverse $φ_i^{-1}(a), i\in I$ of a chart, also called a parametrization.

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Manifolds.induced_basisMethod
induced_basis(M::AbstractManifold, A::AbstractAtlas, i, p, VST::VectorSpaceType)

Basis of vector space of type VST at point p from manifold M induced by chart (A, i).

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Manifolds.local_metricMethod
local_metric(M::AbstractManifold, p, B::InducedBasis)

Compute the local metric tensor for vectors expressed in terms of coordinates in basis B on manifold M. The point p is not checked.

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Manifolds.transition_mapMethod
transition_map(M::AbstractManifold, A_from::AbstractAtlas, i_from, A_to::AbstractAtlas, i_to, a)
transition_map(M::AbstractManifold, A::AbstractAtlas, i_from, i_to, a)

Given coordinates a in chart (A_from, i_from) of a point on manifold M, returns coordinates of that point in chart (A_to, i_to). If A_from and A_to are equal, A_to can be omitted.

Mathematically this function is the transition map or change of charts, but it might even be between two atlases $A_{\text{from}} = \{(U_i,φ_i)\}_{i\in I}$ and $A_{\text{to}} = \{(V_j,\psi_j)\}_{j\in J}$, and hence $I, J$ are their index sets. We have $i_{\text{from}}\in I$, $i_{\text{to}}\in J$.

This method then computes

$$$\bigl(\psi_{i_{\text{to}}}\circ φ_{i_{\text{from}}}^{-1}\bigr)(a)$$$

Note that, similarly to get_parameters, this method should fail the same way if $V_{i_{\text{to}}}\cap U_{i_{\text{from}}}=\emptyset$.

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## Cotangent space and musical isomorphisms

Related to atlases, there is also support for the cotangent space and coefficients of cotangent vectors in bases of the cotangent space.

Functions sharp and flat implement musical isomorphisms for arbitrary vector bundles.

Manifolds.flatMethod
flat(M::AbstractManifold, p, X)

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 AbstractManifold.

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

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Manifolds.sharpMethod
sharp(M::AbstractManifold, p, ξ)

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

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

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