Arch. Math. 109 (2017), 285–292
2017 Springer International Publishing
published online June 1, 2017
Archiv der Mathematik
Projectively induced rotation invariant K¨ahler metrics
Abstract. We classify K¨ahler–Einstein manifolds admitting a K¨ahler
immersion into a ﬁnite dimensional complex projective space endowed
with the Fubini–Study metric, whose codimension is less than or equal to
3 and whose metric is rotation invariant.
Mathematics Subject Classiﬁcation. 53C25, 53C30, 53C55.
Keywords. K¨ahler–Einstein manifolds, Complex projective space, Dias-
1. Introduction. In the present paper we address the problem of studying
holomorphic and isometric (i.e. K¨ahler) immersions of K¨ahler–Einstein (from
now on KE) manifolds into the ﬁnite dimensional complex projective space
endowed with the Fubini–Study metric g
. In particular, we believe the
validity of the following
Conjecture. Every K¨ahler–Einstein manifold which admits a K¨ahler immer-
sion into (CP
),withN<∞, is an open subset of a compact homoge-
An explicit example of non-homogeneous KE manifolds which admit a
K¨ahler immersion into (CP
) can be found in .
The main result of the paper consists in proving the above mentioned con-
jecture in the case of rotation invariant K¨ahler metrics when the codimension
with respect to the target projective space is ≤3. This is shown in the next
section via the following two theorems.
Theorem 1.1. The Einstein constant λ of a KE rotational invariant and pro-
jectively induced n-dimensional manifold M
is a positive rational number less
than or equal to 2(n +1).Hence,ifM
is complete, then M
is compact and