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I wonder if there are some professors only focusing on teaching or management but no research. I think this may be feasible.

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In the Netherlands you can have a "buitengewoon hoogleraar", who has a professor position of typically one day a week and another position somewhere else. Often this is an external entity like a company, but this has also been used to give a professor title (but not the pay) to someone who does the managent of some institute.

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I am an Associate Teaching Professor. My duties include no research. I teach twice as many classes as my colleagues.

A position as a teaching-track faculty member is not uncommon in US universities. But, it isn't common either. My department has about 4% of its faculty on the teaching-track.

Very feasible.

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At my UK university we have a few academic staff whose job title is "Teaching Fellow". These are postdoctoral positions. They spend all of term time teaching, with the chance to work on some of their own research over the summer.

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