Let M ⊂ ℂ² be a Riemann surface that is a holomorphic submanifold of complex 2-space. As such it inherits a Riemannian metric from ℂ² ≈ ℝ⁴.

Each point of M has a tangent "line" — a real 2-plane — which when translated to the origin of ℂ² becomes a complex line in ℂℙ¹ ≈ S², the unit 2-sphere.

This defines a mapping M → S². Now view this as a mapping between real Riemannian surfaces.

Is it true that the Jacobian determinant of this mapping at any p ∈ M is the Gaussian curvature of M at p?

(If not, what is it?)

And are there generalizations to holomorphic curves in ℂⁿ?

Any references to the literature will be appreciated.