A Detailed Study on Attributes, Variable Characterization and Future Consequences in Smooth Maps and Manifolds

Mathematics is an advance science with a special modus operandi. It replaces concrete natural objects with mental abstractions which serve as intermediaries. One studies the properties of these abstractions in the hope they reflect facts of life.

The most visible natural object is the space, the place where all things happen. The first and most important mathematical abstraction is the notion of number. The purpose of this novel study is to illustrate how these two concepts, manifolds and maps fit together. All the other properties of manifolds and maps were derived from these simple axioms.

The concept of smooth manifold through abstract definitions defines the notion of Lie group. The main geometric and algebraic properties of these objects will be gradually described as the geometry of manifolds. Besides their obvious usefulness in geometry, the Lie groups are academically very essential and have a lot of applications in newly defined areas of mathematical sciences. They provide a marvellous testing ground for abstract results.

The reason why smooth manifolds have many differentiable objects attached to them is that they can be locally very well approximated by linear spaces called tangent spaces. Each point has a tangent space attached to it so that we obtain a bunch of tangent spaces called the tangent bundle. Lie groups essentially describing the equivalence between representations and their characters. The study of shape begins in with Riemann manifolds. We approach these objects gradually. The notion of geodesics are defined using the Levi-Civita connection.

Locally, the geodesics play the same role as the straight lines in a Euclidian space but globally new phenomena arise. The study initiates the local study of Riemann manifolds. Up to first order these manifolds look like Euclidian spaces. The novelty arises when we study second order approximations of these spaces. The Riemann tensor provides the complete measure of how far is a Riemann manifold from being flat. This is a very involved object and, to enhance its understanding, we compute it in several instances: on surfaces and on Lie groups.