This article deals with vector valued differential forms on $C^{\infty }$-manifolds. As a generalization of the exterior product, we introduce an operator that combines $Hom({⨂}^s\left(W\right),Z)$-valued forms with $Hom({⨂}^s\left(V\right),W)$-valued forms. We discuss the main properties of this operator such as (multi)linearity, associativity and its behavior under pullbacks, push-outs, exterior differentiation of forms, etc. Finally we present applications for Lie groups and fiber bundles.

We prove a generalization of Thom’s transversality theorem. It gives conditions under which the jet map $f_{*}{|}_Y:Y\subseteq J^r(D,M)\to J^r(D,N)$ is generically (for $f:M\to N$) transverse to a submanifold $Z\subseteq J^r(D,N)$. We apply this to study transversality properties of a restriction of a fixed map $g:M\to P$ to the preimage ${\left(j^{s}f\right)}^{-1}\left(A\right)$ of a submanifold $A\subseteq J^s(M,N)$ in terms of transversality properties of the original map $f$. Our main result is that for a reasonable class of submanifolds $A$ and a generic map $f$ the restriction ${g|}_{{\left(j^{s}f\right)}^{-1}\left(A\right)}$ is also generic. We also present an example of $A$ where the...

Suppose that, for each point x in a given subset E ⊂ Rn, we are given an m-jet f(x) and a convex, symmetric set σ(x) of m-jets at x. We ask whether there exist a function F ∈ Cm,w(Rn) and a finite constant M, such that the m-jet of F at x belongs to f(x) + Mσ(x) for all x ∈ E. We give a necessary and sufficient condition for the existence of such F, M, provided each σ(x) satisfies a condition that we call "Whitnet w-convexity".

The notion of controlled invariance under quasi-static state feedback for discrete-time nonlinear systems has been recently introduced and shown to provide a geometric solution to the dynamic disturbance decoupling problem (DDDP). However, the proof relies heavily on the inversion (structure) algorithm. This paper presents an intrinsic, algorithm-independent, proof of the solvability conditions to the DDDP.

Let $ℳ=(M,{𝒪}_{ℳ})$ be a smooth supermanifold with connection $\nabla$ and Batchelor model ${𝒪}_{ℳ}\cong {\Gamma }_{\Lambda E^{*}}$. From $(ℳ,\nabla )$ we construct a connection on the total space of the vector bundle $E\to M$. This reduction of $\nabla$ is well-defined independently of the isomorphism ${𝒪}_{ℳ}\cong {\Gamma }_{\Lambda E^{*}}$. It erases information, but however it turns out that the natural identification of supercurves in $ℳ$ (as maps from ${ℝ}^{1|1}$ to $ℳ$) with curves in $E$ restricts to a 1 to 1 correspondence on geodesics. This bijection is induced by a natural identification of initial conditions for geodesics...

In this paper, the standard almost complex structure on the tangent bunle of a Riemannian manifold will be generalized. We will generalize the standard one to the new ones such that the induced $(0,2)$-tensor on the tangent bundle using these structures and Liouville $1$-form will be a Riemannian metric. Moreover, under the integrability condition, the curvature operator of the base manifold will be classified.