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We report on our experience formalizing differential geometry with mathlib,the Lean mathematical library. Our account is geared towards geometers with noknowledge of type theory, but eager to learn more about the formalization ofmathematics and maybe curious enough to give Lean a try in the future. To thiseffect, we stress the possibly surprising difference between the formalizationand its pen-and-paper counterpart arising from Lean's treatment of equality.Our three case studies are Lie groups, vector bundles and the Lie algebra of aLie group.

Elements of Differential Geometry in Lean: A Report for Mathematicians