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 ℂ2, i.e., a point of 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.