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Hyperbolic space

Manifolds.Hyperbolic — Type
Hyperbolic{N} <: AbstractEmbeddedManifold{ℝ,DefaultIsometricEmbeddingType}

The hyperbolic space $ℍ^n$ represented by $n+1$-Tuples, i.e. embedded in the Lorentzian manifold equipped with the MinkowskiMetric $⟨\cdot,\cdot⟩_{\mathrm{M}}$. The space is defined as

\[ℍ^n = \Bigl\{p ∈ ℝ^{n+1}\ \Big|\ ⟨p,p⟩_{\mathrm{M}}= -p_{n+1}^2 + \displaystyle\sum_{k=1}^n p_k^2 = -1, p_{n+1} > 0\Bigr\},.\]

The tangent space $T_p ℍ^n$ is given by

\[T_p ℍ^n := \bigl\{ X ∈ ℝ^{n+1} : ⟨p,X⟩_{\mathrm{M}} = 0 \bigr\}.\]

Note that while the MinkowskiMetric renders the Lorentz manifold (only) pseudo-Riemannian, on the tangent bundle of the Hyperbolic space it induces a Riemannian metric. The corresponding sectional curvature is $-1$.

Constructor

Hyperbolic(n)

Generate the $ℍ^{n} ⊂ ℝ^{n+1}$

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

Compute the exponential map on the Hyperbolic space $ℍ^n$ emanating from p towards X. The formula reads

\[\exp_p X = \cosh(\sqrt{⟨X,X⟩_{\mathrm{M}}})p + \sinh(\sqrt{⟨X,X⟩_{\mathrm{M}}})\frac{X}{\sqrt{⟨X,X⟩_{\mathrm{M}}}},\]

where $⟨\cdot,\cdot⟩_{\mathrm{M}}$ denotes the MinkowskiMetric on the embedding, the Lorentzian manifold.

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

Compute the logarithmic map on the Hyperbolic space $ℍ^n$, the tangent vector representing the geodesic starting from p reaches q after time 1. The formula reads for $p ≠ q$

\[\log_p q = d_{ℍ^n}(p,q) \frac{q-⟨p,q⟩_{\mathrm{M}} p}{\lVert q-⟨p,q⟩_{\mathrm{M}} p \rVert_2},\]

where $⟨\cdot,\cdot⟩_{\mathrm{M}}$ denotes the MinkowskiMetric on the embedding, the Lorentzian manifold. For $p=q$ the logarihmic map is equal to the zero vector.

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ManifoldsBase.check_tangent_vector — Method
check_tangent_vector(M::Hyperbolic, p, X; check_base_point = true, kwargs... )

Check whether X is a tangent vector to p on the Hyperbolic M, i.e. after check_manifold_point(M,p), X has to be of the same dimension as p and orthogonal to p with respect to the inner product from the embedding, see MinkowskiMetric. The optional parameter check_base_point indicates whether to call check_manifold_point for p. The tolerance for the last test can be set using the kwargs....

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

Compute the distance on the Hyperbolic M, which reads

\[d_{ℍ^n}(p,q) = \operatorname{acosh}( - ⟨p, q⟩_{\mathrm{M}}),\]

where $⟨\cdot,\cdot⟩_{\mathrm{M}}$ denotes the MinkowskiMetric on the embedding, the Lorentzian manifold.

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

Perform an orthogonal projection with respect to the Minkowski inner product of X onto the tangent space at p of the Hyperbolic space M.

The formula reads

\[Y = X + ⟨p,X⟩_{\mathrm{M}} p,\]

where $⟨\cdot, \cdot⟩_{\mathrm{M}}$ denotes the MinkowskiMetric on the embedding, the Lorentzian manifold.

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ManifoldsBase.vector_transport_to — Method
vector_transport_to(M::Hyperbolic, p, X, q, ::ParallelTransport)

Compute the paralllel transport of the X from the tangent space at p on the Hyperbolic space $ℍ^n$ to the tangent at q along the geodesic connecting p and q. The formula reads

\[\mathcal P_{q←p}X = X - \frac{⟨\log_p q,X⟩_p}{d^2_{ℍ^n}(p,q)} \bigl(\log_p q + \log_qp \bigr),\]

where $⟨\cdot,\cdot⟩_p$ denotes the inner product in the tangent space at p.

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