Universal Bundles of Metrics and Polymetrics: A Generalization

We formalize the ``metric bundle'' viewpoint by defining, for any smooth $n$--manifold $M$, the open fiberwise cones $\mathcal{G}^{p,q}\subset S^2\Tstar M$ of nondegenerate symmetric bilinear forms with fixed signature $(p,q)$, and we package \emph{multi\-metric} (``polymetric'') geometries as sections of finite products of such cones. This framework subsumes Riemannian and pseudo-Riemannian metrics, and admits clean extensions to conformal, densitized, Finsler, and sub-Riemannian structures. It also interfaces correctly with families index theory (Atiyah--Singer), equivariant/groupoid settings, and coarse/KK-theory. On compact $M$ the Riemannian space of metrics is convex (hence contractible), giving a transparent base for moduli and deformation theory