Conformally flat metric $\overline{g}$ is said to be Ricci superosculating with $g$ at the point $x_0$ if $g_{ij}\left(x_0\right)={\overline{g}}_{ij}\left(x_0\right)$, ${\Gamma }_{ij}^k\left(x_0\right)={\overline{\Gamma }}_{ij}^k\left(x_0\right)$, $R_{ij}^k\left(x_0\right)={\overline{R}}_{ij}^k\left(x_0\right)$, where $R_{ij}$ is the Ricci tensor. In this paper the following theorem is proved: ((

In this paper we prove that each $g$-natural metric on a linear frame bundle $LM$ over a Riemannian manifold $(M,g)$ is invariant with respect to a lifted map of a (local) isometry of the base manifold. Then we define $g$-natural metrics on the orthonormal frame bundle $OM$ and we prove the same invariance result as above for $OM$. Hence we see that, over a space $(M,g)$ of constant sectional curvature, the bundle $OM$ with an arbitrary $g$-natural metric $\tilde{G}$ is locally homogeneous.

Let $𝒳\to S$ be a smooth proper family of complex curves (i.e. family of Riemann surfaces), and $ℱ$ a $G$-bundle over $𝒳$ with connection along the fibres $𝒳\to S$. We construct a line bundle with connection $({ℒ}_{ℱ},{\nabla }_{ℱ})$ on $S$ (also in cases when the connection on $ℱ$ has regular singularities). We discuss the resulting $({ℒ}_{ℱ},{\nabla }_{ℱ})$ mainly in the case $G={ℂ}^{*}$. For instance when $S$ is the moduli space of line bundles with connection over a Riemann surface $X$, $𝒳=X\times S$, and $ℱ$ is the Poincaré bundle over $𝒳$, we show that $({ℒ}_{ℱ},{\nabla }_{ℱ})$ provides a prequantization...

Let $F=F^{(A,H,t)}$ and $F^1=F^{(A^1,H^1,t^1)}$ be fiber product preserving bundle functors on the category ${\mathrm{ℱℳ}}_m$ of fibred manifolds $Y$ with $m$-dimensional bases and fibred maps covering local diffeomorphisms. We define a quasi-morphism $(A,H,t)\to (A^1,H^1,t^1)$ to be a $GL\left(m\right)$-invariant algebra homomorphism $\nu :A\to A^1$ with $t^1=\nu \circ t$. The main result is that there exists an ${\mathrm{ℱℳ}}_m$-natural transformation $FY\to F^{1}Y$ depending on a classical linear connection on the base of $Y$ if and only if there exists a quasi-morphism $(A,H,t)\to (A^1,H^1,t^1)$. As applications, we study existence problems of symmetrization (holonomization)...