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They're called flaps and slats.

https://www.grc.nasa.gov/www/k-12/airplane/flap.html

Barnes Wallis developed some interesting designs in the early 50s: http://www.sirbarneswallis.com/Supersonics.htm

Concorde has a variable width wing.

I think they mean variable geometry more in the line of the B-1 Lancer or F-14 Tomcat... And the answer is probably cost, complexity, reliability and weight.

All four of those.