Linearization of Manifolds

Abstract

It is a standard and very useful technique to approximate geometric objects and maps by linear objects and maps. A principal (trivial) example in basic analysis is approximating an open subspace U of \(\mathbb{K}^{n}\) at a point by the linear space \(\mathbb{K}^{n}\) itself by visualizing \(\mathbb{K}^{n}\) as a tangent space to U at the given point. Another (less trivial) example is the derivative of a differentiable map f on U in a point that is the best approximation of f nearby this point by a linear map. In the first section we extend this technique to premanifolds and morphisms of premanifolds. Thus we first define the tangent space of a premanifold at a point, which will be a \(\mathbb{K}\)-vector space that we visualize as the ”best linear approximation“ of the premanifold at the given point. Then we define the derivative in a point x of a morphism f. This will be a linear map from tangent space at x to the tangent space at \(f(x)\).