Aims and outcomes

We shall introduce the concept of an abstract n-dimensional differentiable manifold. This is the natural space where the notion of differentiability of a map can be introduced. The tangent bundle and the subsequent theory of vector fields will be studied in detail. A large part of the course will be devoted to the construction of concrete examples of differentiable manifolds. This will be done by means of different tools ranging from the implicit function theorem up to smooth actions of discrete groups. We will see how algebraic objects such as the linear group or the orthogonal group can be endowed with a natural structure of a differentiable manifold. Meanwhile, we shall present some concepts from the theory of submanifolds and show, according to the (simplified version of a) celebrated theorem by H. Whitney, that every compact abstract manifold can be realised as a smooth subset of some Euclidean space of sufficiently high dimension.

Students are supposed to acquire the fundamental notions from the theory of abstract manifolds and submanifolds with a special attention on different techniques for the construction of concrete examples. We also expect that students acquire the basic theory of vector fields and their flows. Finally, homeworks and exercises should guide the student to develop personal abilities in the abstract investigation of basic differentiable-geometric properties of manifolds.