Metric manifold

AbstractMetric

Abstract type for the pseudo-Riemannian metric tensor $g$, a family of smoothly varying inner products on the tangent space. See inner.

Functor

(metric::Metric)(M::AbstractManifold)
(metric::Metric)(M::MetricManifold)

Generate the MetricManifold that wraps the manifold M with given metric. This works for both a variable containing the metric as well as a subtype T<:AbstractMetric, where a zero parameter constructor T() is availabe. If M is already a metric manifold, the inner manifold with the new metric is returned.

IsDefaultMetric{G<:AbstractMetric}

Specify that a certain AbstractMetric is the default metric for a manifold. This way the corresponding MetricManifold falls back to the default methods of the manifold it decorates.

IsMetricManifold <: AbstractTrait

Specify that a certain decorated Manifold is a metric manifold in the sence that it provides explicit metric properties, extending/changing the default metric properties of a manifold.

MetricManifold{𝔽,M<:AbstractManifold{𝔽},G<:AbstractMetric} <: AbstractDecoratorManifold{𝔽}

Equip a AbstractManifold explicitly with a AbstractMetric G.

For a Metric AbstractManifold, by default, assumes, that you implement the linear form from local_metric in order to evaluate the exponential map.

If the corresponding AbstractMetric G yields closed form formulae for e.g. the exponential map and this is implemented directly (without solving the ode), you can of course still implement that directly.

Constructor

MetricManifold(M, G)

Generate the AbstractManifold M as a manifold with the AbstractMetric G.

RiemannianMetric <: AbstractMetric

Abstract type for Riemannian metrics, a family of positive definite inner products. The positive definite property means that for $X ∈ T_p \mathcal M$, the inner product $g(X, X) > 0$ whenever $X$ is not the zero vector.

log(N::MetricManifold{M,G}, p, q)

Copute the logarithmic map on the AbstractManifold M equipped with the AbstractMetric G.

If the metric was declared the default metric using the IsDefaultMetric trait or is_default_metric, this method falls back to log(M,p,q). Otherwise, you have to provide an implementation for the non-default AbstractMetric G metric within its MetricManifold{M,G}.

change_metric(M::AbstractcManifold, G2::AbstractMetric, p, X)

On the AbstractManifold M with implicitly given metric $g_1$ and a second AbstractMetric $g_2$ this function performs a change of metric in the sense that it returns the tangent vector $Z=BX$ such that the linear map $B$ fulfills

\[g_2(Y_1,Y_2) = g_1(BY_1,BY_2) \quad \text{for all } Y_1, Y_2 ∈ T_p\mathcal M.\]

If both metrics are given in their local_metric (symmetric positive defintie) matrix representations $G_1 = C_1C_1^{\mathrm{H}}$ and $G_2 = C_2C_2^{\mathrm{H}}$, where $C_1,C_2$ denote their respective Cholesky factors, then solving $C_2C_2^{\mathrm{H}} = G_2 = B^{\mathrm{H}}G_1B = B^{\mathrm{H}}C_1C_1^{\mathrm{H}}B$ yields $B = (C_1 \backslash C_2)^{\mathrm{H}}$, where $\cdot^{\mathrm{H}}$ denotes the conjugate transpose.

This function returns Z = BX.

Examples

change_metric(Sphere(2), EuclideanMetric(), p, X)

Since the metric in $T_p\mathbb S^2$ is the Euclidean metric from the embedding restricted to $T_p\mathbb S^2$, this just returns X

change_metric(SymmetricPOsitiveDefinite(3), EuclideanMetric, p, X)

Here, the default metric in $\mathcal P(3)$ is the LinearAffineMetric and the transformation can be computed as $B=p$

change_representer(M::AbstractManifold, G2::AbstractMetric, p, X)

Convert the representer X of a linear function (in other words a cotangent vector at p) in the tangent space at p on the AbstractManifold M given with respect to the AbstractMetric G2 into the representer with respect to the (implicit) metric of M.

In order to convert X into the representer with respect to the (implicitly given) metric $g_1$ of M, we have to find the conversion function $c: T_p\mathcal M \to T_p\mathcal M$ such that

\[ g_2(X,Y) = g_1(c(X),Y)\]

If both metrics are given in their local_metric (symmetric positive defintie) matrix representations $G_1$ and $G_2$ and $x,y$ are the local coordinates with respect to the same basis of the tangent space, the equation reads

\[ x^{\mathrm{H}}G_2y = c(x)^{\mathrm{H}}G_1 y \quad \text{for all } y \in ℝ^d,\]

where $\cdot^{\mathrm{H}}$ denotes the conjugate transpose. We obtain $c(X) = (G_1\backslash G_2)^{\mathrm{H}}X$

For example X could be the gradient $\operatorname{grad}f$ of a real-valued function $f: \mathcal M \to ℝ$, i.e.

\[ g_2(X,Y) = Df(p)[Y] \quad \text{for all } Y ∈ T_p\mathcal M.\]

and we would change the Riesz representer X to the representer with respect to the metric $g_1$.

Examples

change_representer(Sphere(2), EuclideanMetric(), p, X)

Since the metric in $T_p\mathbb S^2$ is the Euclidean metric from the embedding restricted to $T_p\mathbb S^2$, this just returns X

change_representer(SymmetricPositiveDefinite(3), EuclideanMetric(), p, X)

Here, the default metric in $\mathcal P(3)$ is the LinearAffineMetric and the transformation can be computed as $pXp$

connection(::MetricManifold)

Return the LeviCivitaConnection for a metric manifold.

det_local_metric(M::AbstractManifold, p, B::AbstractBasis)

Return the determinant of local matrix representation of the metric tensor $g$, i.e. of the matrix $G(p)$ representing the metric in the tangent space at $p$ with as a matrix.

See also local_metric

einstein_tensor(M::AbstractManifold, p, B::AbstractBasis; backend::AbstractDiffBackend = diff_badefault_differential_backendckend())

Compute the Einstein tensor of the manifold M at the point p, see https://en.wikipedia.org/wiki/Einstein_tensor

flat(N::MetricManifold{M,G}, p, X::TFVector)

Compute the musical isomorphism to transform the tangent vector X from the AbstractManifold M equipped with AbstractMetric G to a cotangent by computing

\[X^♭= G_p X,\]

where $G_p$ is the local matrix representation of G, see local_metric

inverse_local_metric(M::AbstractcManifold{𝔽}, p, B::AbstractBasis)

Return the local matrix representation of the inverse metric (cometric) tensor of the tangent space at p on the AbstractManifold M with respect to the AbstractBasis basis B.

The metric tensor (see local_metric) is usually denoted by $G = (g_{ij}) ∈ 𝔽^{d×d}$, where $d$ is the dimension of the manifold.

Then the inverse local metric is denoted by $G^{-1} = g^{ij}$.

is_default_metric(M::AbstractManifold, G::AbstractMetric)

returns whether an AbstractMetric is the default metric on the manifold M or not. This can be set by defining this function, or setting the IsDefaultMetric trait for an AbstractDecoratorManifold.

local_metric(M::AbstractManifold{𝔽}, p, B::AbstractBasis)

Return the local matrix representation at the point p of the metric tensor $g$ with respect to the AbstractBasis B on the AbstractManifold M. Let $d$denote the dimension of the manifold and $b_1,\ldots,b_d$ the basis vectors. Then the local matrix representation is a matrix $G\in 𝔽^{n\times n}$ whose entries are given by $g_{ij} = g_p(b_i,b_j), i,j\in\{1,…,d\}$.

This yields the property for two tangent vectors (using Einstein summation convention) $X = X^ib_i, Y=Y^ib_i \in T_p\mathcal M$ we get $g_p(X, Y) = g_{ij} X^i Y^j$.

local_metric_jacobian(
    M::AbstractManifold,
    p,
    B::AbstractBasis;
    backend::AbstractDiffBackend,
)

Get partial derivatives of the local metric of M at p in basis B with respect to the coordinates of p, $\frac{∂}{∂ p^k} g_{ij} = g_{ij,k}$. The dimensions of the resulting multi-dimensional array are ordered $(i,j,k)$.

log_local_metric_density(M::AbstractManifold, p, B::AbstractBasis)

Return the natural logarithm of the metric density $ρ$ of M at p, which is given by $ρ = \log \sqrt{|\det [g_{ij}]|}$ for the metric tensor expressed in basis B.

metric(M::MetricManifold)

Get the metric $g$ of the manifold M.

ricci_curvature(M::AbstractManifold, p, B::AbstractBasis; backend::AbstractDiffBackend = default_differential_backend())

Compute the Ricci scalar curvature of the manifold M at the point p using basis B. The curvature is computed as the trace of the Ricci curvature tensor with respect to the metric, that is $R=g^{ij}R_{ij}$ where $R$ is the scalar Ricci curvature at p, $g^{ij}$ is the inverse local metric (see inverse_local_metric) at p and $R_{ij}$ is the Riccie curvature tensor, see ricci_tensor. Both the tensor and inverse local metric are expressed in local coordinates defined by B, and the formula uses the Einstein summation convention.

sharp(N::MetricManifold{M,G}, p, ξ::CoTFVector)

Compute the musical isomorphism to transform the cotangent vector ξ from the AbstractManifold M equipped with AbstractMetric G to a tangent by computing

\[ξ^♯ = G_p^{-1} ξ,\]

where $G_p$ is the local matrix representation of G, i.e. one employs inverse_local_metric here to obtain $G_p^{-1}$.

inner(N::MetricManifold{M,G}, p, X, Y)

Compute the inner product of X and Y from the tangent space at p on the AbstractManifold M using the AbstractMetric G. If M has G as its IsDefaultMetric trait, this is done using inner(M, p, X, Y), otherwise the local_metric(M, p) is employed as

\[g_p(X, Y) = ⟨X, G_p Y⟩,\]

where $G_p$ is the loal matrix representation of the AbstractMetric G.