Two fiber bundles E₁ and E₂ over the same base space M yield the fibered set ℱ(E₁,E₂) → M, whose fibers are defined as $C^{\infty }(E₁ₓ,E₂ₓ)$, for each x ∈ M. This fibered set can be regarded as a smooth space in the sense of Frölicher and we construct its tangent prolongation. Then we extend the Frölicher-Nijenhuis bracket to projectable tangent valued forms on ℱ(E₁,E₂). These forms turn out to be a kind of differential operators. In particular, we consider a general connection on ℱ(E₁,E₂) and study the associated...

In this paper we study Lipschitz-Fredholm vector fields on bounded Fréchet-Finsler manifolds. In this context we generalize the Morse-Sard-Brown theorem, asserting that if $M$ is a connected smooth bounded Fréchet-Finsler manifold endowed with a connection $𝒦$ and if $\xi$ is a smooth Lipschitz-Fredholm vector field on $M$ with respect to $𝒦$ which satisfies condition (WCV), then, for any smooth functional $l$ on $M$ which is associated to $\xi$, the set of the critical values of $l$ is of first category in $ℝ$. Therefore,...

Let $U_c\left(\right)$ = u: u unitary and u-1 compact stand for the unitary Fredholm group. We prove the following convexity result. Denote by $d_{\infty }$ the rectifiable distance induced by the Finsler metric given by the operator norm in $U_c\left(\right)$. If $u₀,u₁,u\in U_c\left(\right)$ and the geodesic β joining u₀ and u₁ in $U_c\left(\right)$ satisfy $d_{\infty }(u,\beta )<\pi /2$, then the map $f\left(s\right)=d_{\infty }(u,\beta \left(s\right))$ is convex for s ∈ [0,1]. In particular, the convexity radius of the geodesic balls in $U_c\left(\right)$ is π/4. The same convexity property holds in the p-Schatten unitary groups $U_p\left(\right)$ = u: u unitary and u-1 in the p-Schatten class...