die Rückkehr
return
The act of coming back to a starting point or home.
Nach langer Reise freute er sich auf seine Rückkehr in die Heimat.
After a long journey, he was looking forward to his return to his homeland.

return
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return
The act of coming back to a starting point or home.
Nach langer Reise freute er sich auf seine Rückkehr in die Heimat.
After a long journey, he was looking forward to his return to his homeland.

'Rückkehr' (return) is a noun derived from the separable verb 'zurückkehren' (to return). It is made up of 'Rück-' (from 'zurück', back) and 'kehr' (from the verb 'kehren', which here means 'to turn'). It describes the action of coming back.
Niemand rechnete mit seiner plötzlichen Rückkehr.
No one expected his sudden return.
The preposition after 'Rückkehr' (return) depends on the destination. You use 'nach' for places without an article (cities, countries), 'in' for places with an article, and 'zu' for states or activities.
Die Rückkehr nach Deutschland war für sie ein großer Schritt.
The return to Germany was a big step for her.
In the figurative meaning, 'Rückkehr' (return) is often used for the restoration of a previous state. Typical phrases are 'die Rückkehr zur Normalität' (the return to normality) or 'die Rückkehr in den Beruf' (the return to work).
Nach der Krise wünschten sich alle eine schnelle Rückkehr zur Normalität.
After the crisis, everyone wanted a quick return to normality.
arrival
'die Rückkehr' and 'die Ankunft' both have to do with reaching a place. The difference lies in the direction. 'die Ankunft' describes someone arriving at a destination. It does not matter whether the person has been there before. 'die Rückkehr' is used when someone comes back to a previous place, for example home or to the workplace. The words can be confused because both can be the end of a trip or journey. If the place is a destination, 'Ankunft' fits. If it is about coming back, 'Rückkehr' fits.

Nach seiner Rückkehr aus dem Urlaub arbeitet Luis wieder im Büro.
After his return from vacation, Luis is working in the office again.
Nach der Ankunft am Bahnhof rief die Fahrerin ihre Familie an.
After arriving at the station, the driver called her family.