Hello, I would like to know if it is possible to make a particular trait not possible to be modified by the identity owner but possible by a user Admin... I can't find in the documentation how to do this. Thank you very much "Traits are the data associated with an identity. This data can be modified by the identity owner, for example at sign up or in the profile update process. Identity traits can also be modified by users with Ory Identities (Kratos) Admin API access."

I think public metadata is what you are looking for

Can we use this public metadata for connexion associated to a password ?

Not sure to understand the question but the public metadata does show up in the

whoami

call, no matter the authentication method used

If I put this into public metadata, the user cannot create his account. When the account is created, I don't want this trait to be modified again

Disclaimer: I’m a complete Ory/Kratos newbie, so take this with a huge grain of salt. Perhaps you can solve this with having two identity schemas: one with this trait available to a user, and another one without it. Make the first one default, so that a user fills it in initially. Use “after registration” hook to move whatever they entered into the public metadata and switch to the second schema.