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Cesàro returned to Italy for a while after the death of his father but returned to Belgium to continue his studies at Liège. He visited Paris and attended lectures by Hermite, Darboux, Serret, Briot, Bouquet and Chasles at the Sorbonne.
He returned to Rome where he wrote 80 works and received a doctorate from the University of Rome in 1887. Cesàro held the chair of mathematics at Palermo until 1891, moving then to Rome where he held the chair until his death.
Cesàro's main contribution was to differential geometry. Influenced by Darboux while in Paris he formulated 'intrinsic geometry'. This is his most important contribution which he described in Lezione di geometria intrinseca (Naples, 1890). This work contains descriptions of curves which today are named after Cesàro. The book also deals with surfaces and n-dimensional spaces.
In addition to differential geometry he worked on many topics such as number theory, divergent series and mathematical physics.
References (6 books/articles)