ABSTRACT
This chapter summarizes some basic tools that are frequently used to design and analyzes the behavior of stochastic processes in constraint type manifolds, including parametric type Riemannian manifolds. It begins with brief discussion on projection operators and symmetric bilinear forms on finite dimensional vector spaces. The chapter explains first and second covariant derivatives of functions and vector fields. It presents more advanced operators such as the divergence, the Lie bracket, the Laplacian, and the Ricci curvature. The chapter is concerned with the Bochner-Lichnerowicz formula and several change-of-variable formulae. The chapter discusses the local expressions of these geometric objects. It also summarizes the linearity properties of the curvature tensor.
