Preliminary Calculus on Manifolds

This chapter provides a preliminary knowledge of manifold. Manifold geometry is the foundation of the geometric approach to dimensionality reduction. In DR, we assume that the observed (high-dimensional) data resides on a low-dimensional manifold whose dimension and shape are not a priori known. A manifold can be represented by its coordinates. DR methods try to find the coordinate representations of data. The current research of differential geometry focuses on the characterization of global properties of manifolds. However, DR methods only need the local properties of manifolds, which is much simpler. In this chapter, we give the notions and notations of manifolds. In Section 1, a linear manifold, also called a hyperplane, is introduced. In Section 2, we discuss differentiable structures of manifolds. An introduction to functions and operators on manifolds is given in Section 3. Most of the material in this chapter can be found in any standard textbook on differential geometry, for example, [1–4]. Therefore, the proofs of most theorems in this chapter are omitted.