Connection manifold

AbstractAffineConnection

Abstract type for affine connections on a manifold.

AbstractConnectionManifold{𝔽,M<:AbstractManifold{𝔽},G<:AbstractAffineConnection} <: AbstractDecoratorManifold{𝔽}

Equip an AbstractManifold explicitly with an AbstractAffineConnection G.

AbstractConnectionManifold is defined by values of christoffel_symbols_second, which is used for a default implementation of exp, log and vector_transport_to. Closed-form formulae for particular connection manifolds may be explicitly implemented when available.

An overview of basic properties of affine connection manifolds can be found in ^[Pennec2020].

ConnectionManifold(M, C)

Decorate the AbstractManifold M with AbstractAffineConnection C.

LeviCivitaConnection

The Levi-Civita connection of a Riemannian manifold.

exp(M::AbstractConnectionManifold, p, X)

Compute the exponential map on the AbstractConnectionManifold M equipped with corresponding affine connection.

If M is a MetricManifold with a metric that was declared the default metric using is_default_metric, this method falls back to exp(M, p, X).

Otherwise it numerically integrates the underlying ODE, see solve_exp_ode. Currently, the numerical integration is only accurate when using a single coordinate chart that covers the entire manifold. This excludes coordinates in an embedded space.

christoffel_symbols_second(
    M::AbstractManifold,
    p,
    B::AbstractBasis;
    backend::AbstractDiffBackend = diff_backend(),
)

Compute the Christoffel symbols of the second kind in local coordinates of basis B. For affine connection manifold the Christoffel symbols need to be explicitly implemented while, for a MetricManifold they are computed as (in Einstein summation convention)

\[Γ^{l}_{ij} = g^{kl} Γ_{ijk},\]

where $Γ_{ijk}$ are the Christoffel symbols of the first kind (see christoffel_symbols_first), and $g^{kl}$ is the inverse of the local representation of the metric tensor. The dimensions of the resulting multi-dimensional array are ordered $(l,i,j)$.

christoffel_symbols_second_jacobian(
    M::AbstractManifold,
    p,
    B::AbstractBasis;
    backend::AbstractDiffBackend = diff_backend(),
)

Get partial derivatives of the Christoffel symbols of the second kind for manifold M at p with respect to the coordinates of B, i.e.

\[\frac{∂}{∂ p^l} Γ^{k}_{ij} = Γ^{k}_{ij,l}.\]

The dimensions of the resulting multi-dimensional array are ordered $(i,j,k,l)$.

connection(M::AbstractManifold)

Get the connection (an object of a subtype of AbstractAffineConnection) of AbstractManifold M.

connection(M::ConnectionManifold)

Return the connection associated with ConnectionManifold M.

gaussian_curvature(M::AbstractManifold, p, B::AbstractBasis; backend::AbstractDiffBackend = diff_backend())

Compute the Gaussian curvature of the manifold M at the point p using basis B. This is equal to half of the scalar Ricci curvature, see ricci_curvature.

ricci_tensor(M::AbstractManifold, p, B::AbstractBasis; backend::AbstractDiffBackend = diff_backend())

Compute the Ricci tensor, also known as the Ricci curvature tensor, of the manifold M at the point p using basis B, see https://en.wikipedia.org/wiki/Ricci_curvature#Introduction_and_local_definition.

riemann_tensor(M::AbstractManifold, p, B::AbstractBasis; backend::AbstractDiffBackend=diff_backend())

Compute the Riemann tensor $R^l_{ijk}$, also known as the Riemann curvature tensor, at the point p in local coordinates defined by B. The dimensions of the resulting multi-dimensional array are ordered $(l,i,j,k)$.

The function uses the coordinate expression involving the second Christoffel symbol, see https://en.wikipedia.org/wiki/Riemann_curvature_tensor#Coordinate_expression for details.

See also

christoffel_symbols_second, christoffel_symbols_second_jacobian

solve_exp_ode(
    M::AbstractConnectionManifold,
    p,
    X,
    tspan,
    B::AbstractBasis;
    backend::AbstractDiffBackend = diff_backend(),
    solver = AutoVern9(Rodas5()),
    kwargs...,
)

Approximate the exponential map on the manifold over the provided timespan assuming the default connection of the given manifold by solving the ordinary differential equation

\[\frac{d^2}{dt^2} p^k + Γ^k_{ij} \frac{d}{dt} p_i \frac{d}{dt} p_j = 0,\]

where $Γ^k_{ij}$ are the Christoffel symbols of the second kind, and the Einstein summation convention is assumed. The arguments tspan and solver follow the OrdinaryDiffEq conventions. kwargs... specify keyword arguments that will be passed to OrdinaryDiffEq.solve.

Currently, the numerical integration is only accurate when using a single coordinate chart that covers the entire manifold. This excludes coordinates in an embedded space.

using OrdinaryDiffEq