# Gaussian curvature of a holomorphic curve in complex 2-space

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.

Just so you'll have the answer with the correct normalizations: Give $$\mathbb{CP}^{n-1}$$, regarded as the space of lines through the origin in $$\mathbb{C}^n$$, its $$\mathrm{U}(n)$$-invariant metric $$h_{n-1}$$, normalized so that a linear $$\mathbb{CP}^1\subset\mathbb{CP}^{n-1}$$ has Gauss curvature $$+1$$. Let $$\gamma:M\to \mathbb{CP}^{n-1}$$ be the Gauss map. Then $$\gamma^*(h_{n-1}) = -2K\,g_M\,,$$ where $$g_M$$ is the metric induced on $$M\subset\mathbb{C}^n$$ and $$K\le 0$$ is the Gauss curvature of $$g_M$$. In particular, note that the Gauss map $$\gamma$$ is (weakly) conformal.
Remark: Note the factor of $$2$$, which is not present for the Gauss map of a minimal surface in $$\mathbb{R}^3$$. Also, note that some sources prefer to normalize $$h_{n-1}$$ so that the area of a linear $$\mathbb{CP}^1\subset\mathbb{CP}^{n-1}$$ is $$\pi$$ rather than $$4\pi$$. In this case, the factor of $$2$$ in the formula above gets replaced by a factor of $$\tfrac12$$.
In any dimension, you can take a Darboux frame $$A_i$$ about $$p$$ (this is an easy effective computation) and a local coordinate about $$p$$ say $$z$$, Then in a neighborhood about the the point, $$C$$ is given by $$p+zA+z^2B+...$$. Now the question becomes computing the curvature of a graph of a function.