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.

Euclidean{T<:Tuple,𝔽} <: Manifold{𝔽}

Euclidean vector space.



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 -, -valued real- or complex-valued immutable values (in contrast to 1-element arrays from the constructor above).

EuclideanMetric <: RiemannianMetric

A general type for any manifold that employs the Euclidean Metric, for example the Euclidean manifold itself, or the Sphere, where every tangent space (as a plane in the embedding) uses this metric (in the embedding).

Since the metric is independent of the field type, this metric is also used for the Hermitian metrics, i.e. metrics that are analogous to the EuclideanMetric but where the field type of the manifold is .

This metric is the default metric for example for the Euclidean manifold.

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

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

log(M::Euclidean, p, q)

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

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

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.

flat(M::Euclidean, p, X)

Transform a tangent vector X into a cotangent. Since they can directly be identified in the Euclidean case, this yields just the identity for a tangent vector w in the tangent space of p on M.

sharp(M::Euclidean, p, ξ)

Transform the cotangent vector ξ at p on the Euclidean M to a tangent vector X. Since cotangent and tangent vectors can directly be identified in the Euclidean case, this yields just the identity.

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.

embed(M::Euclidean, p, X)

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

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.

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.

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.