TITLE: Geometry of complex elliptic curves
QUESTION [23 upvotes]: Is there an elliptic curve in CP^2 whose induced Remannian metric ( induced from the Fubini-Sudy metric on CP^2) is Euclidian flat?

REPLY [6 votes]: 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.