Is there an elliptic curve in CP^2 whose induced Remannian metric ( induced from the Fubini-Sudy metric on CP^2) is Euclidian flat? 



Geometry of complex elliptic curves
According to this [paper](http://www.numdam.org/numdam-bin/item?id=CM_1977__35_1_57_0) by Linda Ness the Gaussian curvature of a curve $C\subset \mathbb P^2$ defined by the 
zeros of a degree $d>1$ homogeneous polynomial $F \in \mathbb C[x,y,z]$ at a smooth
point $p$ is given by
$$
K(p) = 2- \frac{\|p\|^6 \cdot | \rm{Hessian}(F)(p)|^2}{ (d-1)^4 \cdot \| \nabla F(p) \|^6} ,
$$
where $\| \cdot \|$ stands for the usual norm in $\mathbb C^3$, and $\nabla F$ is the gradient of $F$. 


In particular, if $p$ is a smooth inflection point of $C$ then $K(p) = 2$. Thus, there are no smooth cubics in $\mathbb P^2$ which are Euclidean flat, since these have $9$ inflection points. 


N.B. : Ness normalizes the Fubiny-Study metric to have sectional curvature $2$. 




---


After googling a bit I've found the paper [The Riemannian geometry of holomorphic curves](http://www.ams.org/mathscinet-getitem?mr=324606) by Blaine Lawson which is strictly related to the subject. There he says that Eugenio Calabi proved, in [Isometric imbedding of complex manifolds](http://www.ams.org/mathscinet-getitem?mr=57000.), that 



> 
> ($\ldots$) modulo holomorphic congruences, there is only
>  one curve $C\_n$ of
>  constant Gauss curvature in $\mathbb
> > C P^n$ which does not lie in any linear
>  subspace. This curve has curvature $1/n$ and is given by the following embedding of $\mathbb C P^1\to \mathbb C P^n$:
>  $$(z\_0,z\_1) \mapsto \left(z\_0^n, \sqrt{n} z\_0^{n-1} z\_1, \ldots, \sqrt{\binom{n}{k}}z\_0^{n-k}z\_1^k, \ldots, z\_1^n \right). $$
> 
> 
> 


I could not find this statement in Calabi's paper, but this does not exclude the possibility that it is indeed there. The paper is the published version of Calabi's Phd thesis, so another possibility is that the statement is in the thesis but did not make its way into the paper. 


N.B. : Lawson normalizes the Fubiny-Study metric to have sectional curvature $1$.


