About the book
Manifolds, a subject of interest for researchers in their own right, have many applications and interactions with many areas of mathematics and physics. These areas include partial differential equations, elliptic problems, boundary value problems, Schrodinger, and heat operators. Fundamentally, with Descartes and the introduction of coordinates, a line or a plane becomes via coordinates an algebraic object, more precisely an equation.
In general, any coordinates replace geometry by algebra and we get a two-dimensional correspondence between the study of space and the study of equations. This process is a shift from geometry to numbers at a basic level. The coordinatization process has been used well before mathematicians accepted it as a method.
The manifolds are precisely those spaces that can be piecewise provided with coordinates by means of a smooth correspondence on overlaps, and the book will intend to study these structures in mathematics, as well as the impact and applications to a variety of other areas of mathematics. Recently, there have been very deep insights into the subject, and it is intended this the book will provide readers with an interest in the subject a clear review of advances and consequences in this area of investigation.