Inconsistency in invariant metrics

[mateuszbaran]

Currently invariant metrics on groups are somewhat inconsistent: inner continues dispatch with traits without unwrapping the manifold while exp and log unwrap and run trait dispatch from scratch. This should be consistent.

[mateuszbaran]

Related: bases for invariant metrics:

for (invtrait, invdir) in
    [(HasLeftInvariantMetric, LeftAction), (HasRightInvariantMetric, RightAction)],
    basis_type in [DefaultOrthonormalBasis, DefaultOrthogonalBasis]

    eval(
        quote
            function get_coordinates(
                ::TraitList{$invtrait},
                M::AbstractDecoratorManifold,
                p,
                X,
                B::$basis_type,
            )
                Xₑ = inverse_translate_diff(M, p, p, X, $invdir())
                return get_coordinates(M.manifold, p, Xₑ, B)
            end
            function get_coordinates!(
                ::TraitList{$invtrait},
                M::AbstractDecoratorManifold,
                c,
                p,
                X,
                B::$basis_type,
            )
                Xₑ = inverse_translate_diff(M, p, p, X, $invdir())
                return get_coordinates!(M.manifold, c, p, Xₑ, B)
            end

            function get_vector(
                ::TraitList{$invtrait},
                M::AbstractDecoratorManifold,
                p,
                c,
                B::$basis_type,
            )
                Xₑ = get_vector(M.manifold, p, c, B)
                return translate_diff(M, p, p, Xₑ, $invdir())
            end
            function get_vector!(
                ::TraitList{$invtrait},
                M::AbstractDecoratorManifold,
                X,
                p,
                c,
                B::$basis_type,
            )
                Xₑ = get_vector(M.manifold, p, c, B)
                return translate_diff!(M, X, p, p, Xₑ, $invdir())
            end
        end,
    )
end

[mateuszbaran]

I think in general we just shouldn't dispatch on specific traits, and just use them for forwarding of specific functions. This part works reasonably well. Dispatching on traits allows for some nice fallback definitions but it is just too convoluted.