Geometrically, this means that the identity map ϕ : (M,g) → (M,˚g), p ↦→ p, satisfies the inhomogeneous harmonic wave map equation
μ D ˚D ∂ϕA-∂ϕB- μν D ∇ ∇ μϕ + Γ AB ∂xμ ∂xν g = H , (4.2)
where xμ and xA are local coordinates on (M, g) and (M,˚g), respectively.