Question

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

Answer 1

According to this paper 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 $2$.

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After googling a bit I've found the paper The Riemannian geometry of holomorphic curves by Blaine Lawson which is strictly related to the subject. There he says that Eugenio Calabi proved, in Isometric imbedding of complex manifolds, 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 $1$.

Answer 2

A much more general result holds. If $M$ is a compact complex manifold and $f:M\to\mathbb{CP}^n$ is a holomorphic embedding such that $f^*g_{FS}$ is an Einstein metric, then the Einstein constant must be strictly positive. This is a theorem of D. Hulin.

One can also wonder what manifolds $M$ one can obtain in general (i.e. what compact complex submanifolds of $\mathbb{CP}^n$ are Einstein for the induced Fubini-Study metric). It is believed that these must all be complex homogeneous spaces, and a complete classification is known in the case of (complex) codimension at most $2$, and for complete intersections. See this other paper of Hulin and the references there.