Based on the provided source material, there is no information identifying a city named Kleiner in Austria. The source lists "Kleiner" exclusively as a German or Jewish surname meaning "smaller," "lesser," or "minor."
The name appears in various contexts as a surname for individuals of diverse nationalities and professions, including scientists, politicians, actors, and academics. Examples include Swiss physicist Alfred Kleiner, American mathematician Bruce Kleiner, and Israeli politician Michael Kleiner.