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A manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n.

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Background to learn about manifolds

For a more mathematical purely formal treatment of differential geometry on manifolds I would dive in the wonderful book by Nicolaescu - "Lectures on the Geometry of Manifolds" since it is very complete … For an approach to manifolds through mechanics, the classic book of V.I. …
Javier Álvarez's user avatar