What are the responsibilities of a project manager in a DevOps project?
Truth would be that should not happen in a DevOps environment
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Your "project" will probably use Agile concepts, but there's nothing in DevOps telling how a project has to be run because DevOps is product centric (more details on the blog post and with search on this term).

"DevOps" is about breaking silos and have a team composed of developers, architects, ops, exploitation peoples who handle a software on its lifetime, that is antagonistic with the waterfall project approach.

To quote here Jez Humble:


  Products not projects


So your answer would be : There's not responsibilities for a project manager on a DevOps product because the main idea behind devops is to have a team handling the product on long term and not on a planned project time only.