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.

Let $M\subset\mathbb C^2$ be a Riemann surface that is a holomorphic submanifold of complex 2-space. As such it inherits a Riemannian metric from $\mathbb C^2\approx\mathbb R^4$.

Each point of $M$ has a tangent "line" — a real 2-plane — which when translated to the origin of $\mathbb C^2$ becomes a complex line in $\mathbb C^2$, i.e., a point of $\mathbb{CP}^1\approx\mathbf S^2$, the unit 2-sphere.

This defines a mapping $M \to \mathbf S^2$. Now view this as a mapping between real Riemannian surfaces.

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

(If not, what is it?)

And are there generalizations to holomorphic curves in $\mathbb C^n$?

Any references to the literature will be appreciated.

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.

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.