Original manuscript: 2020/09/20
Manuscript last revised: 2022/02/14

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.

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