I do not know much differential geometry, and was led to this question from complex dynamics. It seems that it is often possible to reason 'infinitesimally' about maps between tangent spaces. For example, quasiconformal maps are typically motivated and thought of as sending infinitesimal circles to infinitesimal ellipses, but then defined via pullbacks of almost-complex structures. Why is this possible? What is the correspondence between 'classical' infinitesimals and maps on tangent spaces? Put differently, why should "infinitesimal = lives in tangent space"?
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Without going into much detail, manifolds have the locally euclidean property which means that in a neighbourhood of a point there is an open set which looks like (is homeomorphic to) an open set in $\mathbb{R}^n$. The idea of infintesimals living in the (co)tangent space is basically that if I zoom into the open neighbourhood the set starts looking like $\mathbb{R}^n$ which is a vector space, so I can sort of make sense of infintesimals living in a vector space. This motivates us to define the tangent space more formally using curves and so on.
