Connection manifold

AbstractAffineConnection

Abstract type for affine connections on a manifold.

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

Constructor

ConnectionManifold(M, C)

Decorate the AbstractManifold M with AbstractAffineConnection C.

IsConnectionManifold <: AbstractTrait

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

IsDefaultConnection{G<:AbstractAffineConnection}

Specify that a certain AbstractAffineConnection is the default connection for a manifold. This way the corresponding ConnectionManifold falls back to the default methods of the manifold it decorates.

LeviCivitaConnection

The Levi-Civita connection of a Riemannian manifold.

exp(::TraitList{IsConnectionManifold}, M::AbstractDecoratorManifold, p, X)

Compute the exponential map on a manifold that IsConnectionManifold M equipped with corresponding affine connection.

If M is a MetricManifold with a IsDefaultMetric trait, 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_first(
    M::AbstractManifold,
    p,
    B::AbstractBasis;
    backend::AbstractDiffBackend = default_differential_backend(),
)

Compute the Christoffel symbols of the first kind in local coordinates of basis B. The Christoffel symbols are (in Einstein summation convention)

\[Γ_{ijk} = \frac{1}{2} \Bigl[g_{kj,i} + g_{ik,j} - g_{ij,k}\Bigr],\]

where $g_{ij,k}=\frac{∂}{∂ p^k} g_{ij}$ is the coordinate derivative of the local representation of the metric tensor. The dimensions of the resulting multi-dimensional array are ordered $(i,j,k)$.

christoffel_symbols_second(
    M::AbstractManifold,
    p,
    B::AbstractBasis;
    backend::AbstractDiffBackend = default_differential_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 = default_differential_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 = default_differential_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.

is_default_connection(M::AbstractManifold, G::AbstractAffineConnection)

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

ricci_tensor(M::AbstractManifold, p, B::AbstractBasis; backend::AbstractDiffBackend = default_differential_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.

solve_exp_ode(
    M::ConnectionManifold,
    p,
    X,
    t::Number;
    B::AbstractBasis = DefaultOrthonormalBasis(),
    backend::AbstractDiffBackend = default_differential_backend(),
    solver = AutoVern9(Rodas5()),
    kwargs...,
)

Approximate the exponential map on the manifold by evaluating the ODE descripting the geodesic at 1, 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 argument solver follows 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

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