  The continuity, in a suitable topology, of algebraic and geometric operations
on real analytic manifolds and vector bundles is proved. This is carried out
using recently arrived at seminorms for the real analytic topology. A new
characterisation of the topology of the space of real analytic mappings between
manifolds is also developed. To characterise these topologies, geometric
decompositions of various jet bundles are given by use of connections. These
decompositions are then used to characterise many of the standard operations
from differential geometry: algebraic operations, tensor evaluation, various
lifts of tensor fields, compositions of mappings, etc. Apart from the main
results, numerous techniques are developed that will facilitate the performing
of analysis on real analytic manifolds.
