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Euclidean space

The Euclidean space $ℝ^n$ is a simple model space, since it has curvature constantly zero everywhere; hence, nearly all operations simplify. The easiest way to generate an Euclidean space is to use a field, i.e. AbstractNumbers, e.g. to create the $ℝ^n$ or $ℝ^{n\times n}$ you can simply type M = ℝ^n or ℝ^(n,n), respectively.

Manifolds.Euclidean — Type
Euclidean{T<:Tuple,𝔽} <: AbstractManifold{𝔽}

Euclidean vector space.

Constructor

Euclidean(n)

Generate the $n$-dimensional vector space $ℝ^n$.

Euclidean(n₁,n₂,...,nᵢ; field=ℝ)
𝔽^(n₁,n₂,...,nᵢ) = Euclidean(n₁,n₂,...,nᵢ; field=𝔽)

Generate the vector space of $k = n_1 \cdot n_2 \cdot … \cdot n_i$ values, i.e. the manifold $𝔽^{n_1, n_2, …, n_i}$, $𝔽\in\{ℝ,ℂ\}$, whose elements are interpreted as $n_1 × n_2 × … × n_i$ arrays. For $i=2$ we obtain a matrix space. The default field=ℝ can also be set to field=ℂ. The dimension of this space is $k \dim_ℝ 𝔽$, where $\dim_ℝ 𝔽$ is the real_dimension of the field $𝔽$.

Euclidean(; field=ℝ)

Generate the 1D Euclidean manifold for an ℝ-, ℂ-valued real- or complex-valued immutable values (in contrast to 1-element arrays from the constructor above).

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Base.exp — Method
exp(M::Euclidean, p, X)

Compute the exponential map on the Euclidean manifold M from p in direction X, which in this case is just

\[\exp_p X = p + X.\]

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Base.log — Method
log(M::Euclidean, p, q)

Compute the logarithmic map on the Euclidean M from p to q, which in this case is just

\[\log_p q = q-p.\]

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LinearAlgebra.norm — Method
norm(M::Euclidean, p, X)

Compute the norm of a tangent vector X at p on the Euclidean M, i.e. since every tangent space can be identified with M itself in this case, just the (Frobenius) norm of X.

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ManifoldsBase.distance — Method
distance(M::Euclidean, p, q)

Compute the Euclidean distance between two points on the Euclidean manifold M, i.e. for vectors it's just the norm of the difference, for matrices and higher order arrays, the matrix and ternsor Frobenius norm, respectively.

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ManifoldsBase.embed — Method
embed(M::Euclidean, p, X)

Embed the tangent vector X at point p in M. Equivalent to an identity map.

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ManifoldsBase.inner — Method
inner(M::Euclidean, p, X, Y)

Compute the inner product on the Euclidean M, which is just the inner product on the real-valued or complex valued vector space of arrays (or tensors) of size $n_1 × n_2 × … × n_i$, i.e.

\[g_p(X,Y) = \sum_{k ∈ I} \overline{X}_{k} Y_{k},\]

where $I$ is the set of vectors $k ∈ ℕ^i$, such that for all

$i ≤ j ≤ i$ it holds $1 ≤ k_j ≤ n_j$ and $\overline{\cdot}$ denotes the complex conjugate.

For the special case of $i ≤ 2$, i.e. matrices and vectors, this simplifies to

\[g_p(X,Y) = X^{\mathrm{H}}Y,\]

where $\cdot^{\mathrm{H}}$ denotes the Hermitian, i.e. complex conjugate transposed.

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ManifoldsBase.project — Method
project(M::Euclidean, p, X)

Project an arbitrary vector X into the tangent space of a point p on the Euclidean M, which is just the identity, since any tangent space of M can be identified with all of M.

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ManifoldsBase.vector_transport_to — Method
vector_transport_to(M::Euclidean, p, X, q, ::AbstractVectorTransportMethod)

Transport the vector X from the tangent space at p to the tangent space at q on the Euclidean M, which simplifies to the identity.

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