Integration

manifold_volume(M::AbstractManifold)

Volume of manifold M defined through integration of Riemannian volume element in a chart. Note that for many manifolds there is no universal agreement over the exact ranges over which the integration should happen. For details see [BST03].

volume_density(M::AbstractManifold, p, X)

Volume density function of manifold M, i.e. determinant of the differential of exponential map exp(M, p, X). Determinant can be understood as computed in a basis, from the matrix of the linear operator said differential corresponds to. Details are available in Section 4.1 of [CLLD22].

Note that volume density is well-defined only for X for which exp(M, p, X) is injective.