  A Riemannian metric bundle G(M) is a fiber bundle over a smooth manifold M,
whose fibers are the spaces of symmetric, positive-definite bilinear forms on
the tangent spaces of M, which represent the Rieman?nian metrics. In this work,
we aim to study the category of Riemannian metric bundles and explore their
connections with K-theory and other areas of mathematics. Our main motivation
comes from the idea of multi-norms in Banach spaces, which have found
applications in di?verse fields such as functional analysis, geometric group
theory, and noncommutative geometry. The novelty of our work lies in the
rigorous development of the theory of Riemannian metric bundles, and the
application of this theory to the study of K-theory and other geometric
invariants of manifolds. We hope that our work will contribute to a deeper
understanding of the geometry and topology of manifolds equipped with
Riemannian metric bundles, and provide new insights into the interplay between
geometry, topology, and analysis.
