The curvature tensor is defined via the commutators of second covariant derivatives acting on tensor densities. It is shown that curvature is responsible for the path-dependence of parallel transport. Algebraic and differential identities obeyed by the curvature tensor are derived. The geodesic deviation is defined, and the equation governing it is derived.

Descriptions of the inhomogeneity including dislocations and defects based on the differential geometry forms the basic core of FTMP. This chapter first provides the basic notions of differential geometry necessary for understanding “non-Riemannian plasticity.” The fundamental concepts and quantities are presented second, which is followed by some new features peculiar to the present field theory of multiscale plasticity.

In this chapter we introduce "Ehresmann connections,"with their associated notions of parallel transportand curvature. We then specify these notions in thecase of a Riemannian manifold, where one can find acanonical connection associated with the metricstructure, called Levi–Civita connection. We thenexplain how this connection is related to the theoryof Jacobi curves developed in the previouschapter.