Let M ⊂ ℂ<sup>2</sup> be a Riemann surface that is a holomorphic submanifold of complex 2-space. As such it inherits a Riemannian metric from ℂ<sup>2</sup> ≈ ℝ<sup>4</sup>.

Each point of M has a tangent "line" — a real 2-plane — which when translated to the origin of ℂ<sup>2</sup> becomes a complex line in ℂ<sup>2</sup>, i.e., a point of ℂℙ<sup>1</sup> ≈ **S**<sup>2</sup>, the unit 2-sphere.

This defines a mapping M → **S**<sup>2</sup>. 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 ℂ<sup>n</sup>?

Any references to the literature will be appreciated.