In this note we propose a new way of constructing compact 7-manifolds with a closed G 2-structure. As a result we find a first example of a closed G 2-structure on S 3 × S 4. We also prove that any integral closed G 2-structure on a compact 7-manifold M 7 can be obtained by embedding M 7 to a universal space (W 3(80+8.C 3 8) , h). MSC: 53C10, 53C42 Key words: closed G 2-structures, submanifolds...

We revisit the pioneering work of Bressan & Hong on deterministic control problems in stratified domains, i.e. control problems for which the dynamic and the cost may have discontinuities on submanifolds of R . By using slightly different methods, involving more partial differential equations arguments, we (i) slightly improve the assumptions on the dynamic and the cost; (ii) obtain a compariso...

This is an introduction to the subject of the differential topology of the space of smooth loops in a finite dimensional manifold. It began as background notes to a series of seminars given at NTNU and subsequently at Sheffield. The topics covered are: the smooth structure of the space of smooth loops; constructions involving vector bundles; submanifolds and tubular neighbourhoods; and a short ...

We consider the space, CR(M), consisting of CR functions which also lie in L(M) on a quadric submanifold M of C of codimension at least one. For 1 ≤ p ≤ ∞, we prove that each element in CR(M) extends uniquely to an H function on the interior of the convex hull of M . As part of the proof, we establish a semi-global version of the CR approximation theorem of Baouendi and Treves for submanifolds ...

We study a large class of Poisson manifolds, derived from Manin triples, for which we construct explicit partitions into regular Poisson submanifolds by intersecting certain group orbits. Examples include all varieties L of Lagrangian subalgebras of reductive quadratic Lie algebras d with Poisson structures defined by Lagrangian splittings of d. In the special case of g ⊕ g, where g is a comple...

In this article we obtain the pinching of the null sectional curvature of CRlightlike submanifolds of an inde…nite Hermitian manifold. As a result of this inequality we conclude some non-existence results of such lightlike submanifolds.Moreover using the Index form we prove more non-existence results for CR-lightlike submanifolds. 1. Introduction The theory of submanifolds of a Riemannian or se...

We study a class of Poisson manifolds for which intersections of certain group orbits give partitions into regular Poisson submanifolds. Examples are the varieties L of Lagrangian subalgebras of reductive quadratic Lie algebras with the Poisson structures defined by Lagrangian splittings. In the special case of g⊕g, where g is a complex semi-simple Lie algebra, we explicitly compute the ranks o...

We study a class of Poisson manifolds for which intersections of certain group orbits give partitions into regular Poisson submanifolds. Examples are the varieties L of Lagrangian subalgebras of reductive quadratic Lie algebras with Poisson structures defined by Lagrangian splittings. In the special case of g ⊕ g, where g is a complex semi-simple Lie algebra, we explicitly compute the ranks of ...

A submanifold of a Euclidean space is said to be of constant-ratio if the ratio of the length of the tangential and normal components of its position vector function is constant. The notion of constant-ratio submanifolds was first introduced and studied by the author in [5, 8] during the early 2000s. Such submanifolds relate to a problem in physics concerning the motion in a central force field...

Lagrangian //-umbilical submanifolds are the "simplest" Lagrangian submanifolds next to totally geodesic ones in complex-space-forms. The class of Lagrangian //-umbilical submanifolds in complex Euclidean spaces includes Whitney's spheres and Lagrangian pseudo-spheres. For each submanifold M of Euclidean «-space and each unit speed curve F in the complex plane, we introduce the notion of the co...