  We introduce analogues of the Fubini-Study metrics and the corresponding
Levi-Civita connections on quantum projective spaces. We define the quantum
metrics as two-tensors, symmetric in the appropriate sense, in terms of the
differential calculi introduced by Heckenberger and Kolb. We define connections
on these calculi and show that they are torsion free and cotorsion free, where
the latter condition uses the quantum metric and is a weaker notion of metric
compatibility. Finally we show that these connections are bimodule connections
and that the metric compatibility also holds in a stronger sense.
