Lecturers

Abstract

Smooth manifolds are found everywhere: they appear in several branches of mathematics (beginning at the most elementary level with curves and surfaces), in theoretical Physics (especially in mechanics), and in many scientific and technical applications of Mathematics. Smooth manifolds are spaces locally resembling Euclidean space in which differential calculus may be performed. This calculus can be done by means of co-ordinates, but must not depend on the coordinates employed (we say that it must be intrinsic or geometric). That is why it is necessary to construct a theory in order to be able to work directly with geometric concepts. This course is an introduction to smooth manifolds and is a basic grounding for more advanced studies in pure mathematics (such as Riemann and symplectic geometry) or applied mathematics (such as mechanics or control theory). The course goals in greater detail are as follows: * To learn and fully understand the basic concepts: smooth manifold, differentiable map, tangent and cotangent spaces, tangent map, submanifolds, vector fields and differential 1-forms, tensor fields, etc. * To learn calculus with the above-mentioned objects, both in co-ordinates and intrinsically. * To understand the geometric interpretation of the objects studied and relate them to those studied previously on th