A submanifold is called tangentially degenerate if its Gauss mapping is degenerate. The investigation of tangentially degeneracy of submanifolds has long history. For example the classification of surfaces in R with degenerate Gauss mapping is equivalent to the classification of flat surfaces in R. As a result, that is one of planes, cylinders, cones or tangent developable surfaces. In this pap...

Given a control-affine system and a controlled invariant submanifold, we present necessary and sufficient conditions for local feedback equivalence to a system whose dynamics transversal to the submanifold are linear and controllable. A key ingredient used in the analysis is the new notion of transverse controllability indices of a control system with respect to a set.

Let G be a definably compact definable C group and 1 ≤ r < ∞. Let X be a definable CG submanifold of a representation of G and Y a definable C submanifold of R. We prove that every G invariant surjective submersive definable C map f : X → Y is piecewise definably CG trivial.

By applying the spectral decomposition of a submanifold of a Euclidean space, we derive several sharp geometric inequalities which provide us some best possible relations between volume, center of mass, circumscribed radius, inscribed radius, order, and mean curvature of the submanifold. Several of our results sharpen some well-known geometric inequalities.

A Hamiltonian stationary Lagrangian submanifold of a Kähler manifold is a Lagrangian submanifold whose volume is stationary under Hamiltonian variations. We find a sufficient condition on the curvature of a Kähler manifold of real dimension four to guarantee the existence of a family of small Hamiltonian stationary Lagrangian tori. Mathematics Subject Classification (2000) 58J37 · 35J20 · 35J48...

For an element $\Psi$ in the graded vector space $\Omega^*(M, TM)$ of tangent bundle valued forms on a smooth manifold $M$, $\Psi$-submanifold is defined as submanifold $N$ $M$ such that $\Psi_{|N} \in \Omega^*(N, TN)$. The class $\Psi$-submanifolds encompasses calibrated submanifolds, complex submanifolds and all Lie subgroups compact groups. carries natural algebra structure, given by Frolich...

where Xo is a constant vector and AX,-t — XitX{t, t = 1, 2, . . . , k. If in particular all eigenvalues {Atl, . . . , A,t} are mutually different, then M is said to be of i-type. A Jfc-type submanifold is said to be null if one of the A;t, t — 1, 2, . . . , k, is null. It is easy to see that if M is compact, then Xo in (1.1) is exactly the centre of mass in E . A submanifold M of a hypersphere ...

In analogy with classical submanifold theory, we introduce morphisms of real metric calculi together noncommutative embeddings. We show that basic concepts, such as the second fundamental form and Weingarten map, translate into setting and, in particular, prove a analogue Gauss’ equations for curvature submanifold. Moreover, mean an embedding is readily introduced, giving natural definition min...