ABSTRACT
This chapter introduces the notions of connection and covariant derivative on a general manifold. These additional structures allow us to formulate tensor field equations on a manifold in an invariant way. There is a natural connection on every geometric manifold, the Levi-Civita connection. The chapter lists some of the basic properties of the directional covariant derivative. The covariant derivative not only provides an invariant differentiation measure on a geometric manifold, but it can be employed also to “parallel transport” a vector from one point to another point without altering that vector. The chapter considers the notion of parallel transport of a vector along a curve. It is particularly fruitful to consider the parallel transport of the tangent vector of a curve along that curve. The condition for parallel displacement can be brought into a form more suited for practical applications.
