Metric manifold

A Riemannian manifold always consists of a topological manifold together with a smoothly varying metric $g$.

However, often there is an implicitly assumed (default) metric, like the usual inner product on Euclidean space. This decorator takes this into account. It is not necessary to use this decorator if you implement just one (or the first) metric. If you later introduce a second, the old (first) metric can be used with the (non MetricManifold) AbstractManifold, i.e. without an explicitly stated metric.

This manifold decorator serves two purposes:

1. to implement different metrics (e.g. in closed form) for one AbstractManifold
2. to provide a way to compute geodesics on manifolds, where this AbstractMetric does not yield closed formula.

Note that a metric manifold is has a IsConnectionManifold trait referring to the LeviCivitaConnection of the metric $g$, and thus a large part of metric manifold's functionality relies on this.

Let's first look at the provided types.

Types

Manifolds.AbstractMetricType
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.

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Manifolds.IsMetricManifoldType
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.

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Manifolds.MetricManifoldType
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.

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Manifolds.RiemannianMetricType
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.

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Implement Different Metrics on the same Manifold

In order to distinguish different metrics on one manifold, one can introduce two AbstractMetrics and use this type to dispatch on the metric, see SymmetricPositiveDefinite. To avoid overhead, one AbstractMetric can then be marked as being the default, i.e. the one that is used, when no MetricManifold decorator is present. This avoids reimplementation of the first existing metric, access to the metric-dependent functions that were implemented using the undecorated manifold, as well as the transparent fallback of the corresponding MetricManifold with default metric to the undecorated implementations. This does not cause any runtime overhead. Introducing a default AbstractMetric serves a better readability of the code when working with different metrics.

Implementation of Metrics

For the case that a local_metric is implemented as a bilinear form that is positive definite, the following further functions are provided, unless the corresponding AbstractMetric is marked as default – then the fallbacks mentioned in the last section are used for e.g. the exponential map.

Base.logMethod
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}.

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Manifolds.change_metricMethod
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 $B=p$

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Manifolds.change_representerMethod
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 $pXp$

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Manifolds.inverse_local_metricMethod
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 $G^{-1} = g^{ij}$.

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Manifolds.local_metricMethod
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 $g_p(X, Y) = g_{ij} X^i Y^j$.

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Manifolds.local_metric_jacobianMethod
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 $(i,j,k)$.

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Manifolds.log_local_metric_densityMethod
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.

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Manifolds.ricci_curvatureMethod
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.

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Manifolds.sharpMethod
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 $G_p^{-1}$.

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ManifoldsBase.innerMethod
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.

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Metrics, charts and bases of vector spaces

Metric-related functions, similarly to connection-related functions, need to operate in a basis of a vector space, see here.

Metric-related functions can take bases of associated tangent spaces as arguments. For example local_metric can take the basis of the tangent space it is supposed to operate on instead of a custom basis of the space of symmetric bilinear operators.