ABSTRACT
The basic motivation for the concept of a manifold is to be able to talk about differentiability of functions defined on such objects and then to be able to assign a meaning to the derivative of a differentiable function on such objects. In this chapter, first we introduce the basic concept of a manifold, though we shall restrict ourselves to those that are subspaces of Euclidean spaces. Next, we introduce the concept of tangent space and define the derivative of a differentiable function. We then introduce the reader to special types of maps such as immersions, submersions embeddings etc. In particular, we shall see how the concept of regularity is generalized to transversality. Finally, the concept of “perturbations” is realized in the precise form of a homotopy and some of the special types of maps above are shown to be “stable under small perturbations”. This makes the study of these special maps more important.
