I am trying to connect to a cloud SQL database from cloud composer with the private IP. However, it doesn't work.
I have tried with a self-managed instance and with a cloud SQL with a public IP and it works fine.
Does anyone have a clue to achieve this ?
To connect to a CloudSQL Private IP instance, you could create a Private IP Composer Environment and then deploy a SQL proxy pod in the Composer's underlying GKE cluster and expose this service using ClusterIP.
Here's the Composer documentation where you can find more information regarding this process.