"He sleeps in the afternoon."
Translation:Han sover om eftermiddagen.
When indicating a time of the day, or a day in the week. "Han sover OM torsdagen" - "Han sover OM aftenen"
Also means 'about'. "Hvad taler i OM?" = "What are you talking about?"
For now, a PDF of Danish: A Practical Grammar by Allan, Holmes and Lundskaer-Nielsen is available on the web.
In the year 2000 edition, "Om" the preposition is discussed at paragraph 121, paper page 125, PDF page 140 of 225.
There are examples of "om" as a preposition of:
- time (in the morning, on Fridays, twice a day, twice per week)
- location surrounding (around the neck)
- subject matter (a book about horses, a dissertation on Locke)
"Om" is also a conjunction ( if - when there is uncertainty or doubt expressed).
Some pages of Danish: A Comprehensive Grammar, the 2015 edition, can be viewed as a google books preview. However, "om" the preposition is discussed mostly on pages that are not available in the preview.