There are three manifolds: Grassmann, Stiefel and Unitary, corresponding to submodules of TensorKitManifolds, whose names are exported.

They all have a function Δ = project(!)(X,W) (e.g. Grassmann.project(!) etc) to project an arbitrary tensor X onto the tangent space of W, which is assumed to be isometric/unitary (not checked). The exclamation mark denotes that X is destroyed in the process. The result Δ is of a specific type, the corresponding TensorMap object can be obtained via an argumentless getindex, i.e. Δ[] returns the corresponding TensorMap. However, you typically don't need those. The base point W is also stored in Δ and can be returned using W = base(Δ). Hence, Δ should be assumed to be a point (W, Δ[]) on the tangent bundle of the manifold.

The objects Δ returned by project(!) also satisfy the behaviour of vector: they have scalar multiplication, addition, left and right in-place multiplication with scalars using lmul! and rmul!, axpy! and axpby! as well as complex euclidean inner product dot and corresponding norm. When combining two tangent vectors using addition or inner product, they need to have the same base.

Furthermore, there are the routines required for OptimKit.jl, which also directly work with the objects returned by project(!):

• W′, Δ′ = retract(W, Δ, α): retract W in the direction of Δ with step length α, return both the retracted isometry W′ as well as the local tangent Δ′
• inner(W, Δ₁, Δ₂): inner product between tangent vectors at the point W. Note that W is already encoded in base(Δ₁) and base(Δ₂), but this is the required interface for the inner product of OptimKit.jl. inner(W, Δ₁, Δ₂; metric = :euclidean) = real(dot(Δ₁,Δ₂)) but other metrics might also be available.
• Θ′ = transport(!)(Θ, W, Δ, α, W′): transport tangent vector Θ along the retraction of W in the direction of Δ with step length α, which ends at W′. The result is a the transported vector Θ′ with base(Θ′) == W′. The method with exclamation mark destroys Θ in the process.

When multiple methods are avaible, they are specified using a keyword argument to the above methods, or explicitly as Stiefel.inner_euclidean, Stiefel.inner_canonical, Stiefel.project_euclidean(!), Stiefel.project_canonical(!), Stiefel.retract_exp, Stiefel.transport_exp(!), Stiefel.retract_cayley, Stiefel.transport_cayley(!), Unitary.transport_parallel(!), Unitary.transport_stiefel(!).