General method for including Stueckelberg fields

A systematic procedure is proposed for inclusion of Stueckelberg fields. The procedure begins with the involutive closure when the original Lagrangian equations are complemented by all the lower order consequences. The Stueckelberg field is introduced for every consequence included into the closure. The generators of the Stueckelberg gauge symmetry begin with the operators generating the closure of original system. These operators are not assumed to be a generators of gauge symmetry of any part of the original action, nor are they supposed to form an on shell integrable distribution. With the most general closure generators, the consistent gauge invariant theory is iteratively constructed, without obstructions at any stage. The Batalin–Vilkovisky form of inclusion of the Stueckelberg fields is worked out and the existence theorem for the Stueckelberg action is proven.

First Integrals of Differential Operators from SL(2,ℝ) Symmetries

The construction of first integrals for SL(2,R)-invariant nth-order ordinary differential equations is a non-trivial problem due to the nonsolvability of the underlying symmetry algebra sl(2,R). Firstly, we provide for n=2 an explicit expression for two non-constant first integrals through algebraic operations involving the symmetry generators of sl(2,R), and without any kind of integration. Moreover, although there are cases when the two first integrals are functionally independent, it is proved that a second functionally independent first integral arises by a single quadrature. This result is extended for n>2, provided that a solvable structure for an integrable distribution generated by the differential operator associated to the equation and one of the prolonged symmetry generators of sl(2,R) is known. Several examples illustrate the procedures.

Torsion and the second fundamental form for distributions

The second fundamental form of Riemannian geometry is generalised to the case of a manifold with a linear connection and an integrable distribution. This bilinear form is generally not symmetric and its skew part is the torsion. The form itself is closely related to the shape map of the connection. The codimension one case generalises the traditional shape operator of Riemannian geometry.

DEFORMATIONS OF LEVI FLAT STRUCTURES IN SMOOTH MANIFOLDS

We study intrinsic deformations of Levi flat structures on a smooth manifold. A Levi flat structure on a smooth manifold L is a couple (ξ, J) where ξ ⊂ T(L) is an integrable distribution of codimension 1 and J : ξ → ξ is a bundle automorphism which defines a complex integrable structure on each leaf. A deformation of a Levi flat structure (ξ, J) is a smooth family {(ξt, Jt)}t∈]-ε,ε[ of Levi flat structures on L such that (ξ0, J0) = (ξ, J). We define a complex whose cohomology group of order 1 contains the infinitesimal deformations of a Levi flat structure. In the case of real analytic Levi flat structures, this cohomology group is [Formula: see text] where (𝒵*(L), δ, {⋅,⋅}) is the differential graded Lie algebra associated to ξ.

On Lightlike Geometry of Para-Sasakian Manifolds

We study lightlike hypersurfaces of para-Sasakian manifolds tangent to the characteristic vector field. In particular, we define invariant lightlike hypersurfaces and screen semi-invariant lightlike hypersurfaces, respectively, and give examples. Integrability conditions for the distributions on a screen semi-invariant lightlike hypersurface of para-Sasakian manifolds are investigated. We obtain a para-Sasakian structure on the leaves of an integrable distribution of a screen semi-invariant lightlike hypersurface.

On lightlike geometry in indefinite Kenmotsu manifolds

We investigate some geometric aspects of lightlike hypersurfaces of indefinite Kenmotsu manifolds, tangent to the structure vector field, by paying attention to the geometry of leaves of integrable distributions. Theorems on parallel vector fields, Killing distribution, geodesibility of their leaves are obtained. The geometric configuration of such lightlike hypersurfaces and leaves of its screen integrable distributions are established. We show that no totally contact umbilical leaf of a screen integrable distribution of a lightlike hypersurface can be an extrinsic sphere. We also prove that the geometry of any leaf of an integrable distribution is closely related to the geometry of a normal bundle.

Convolutions with the Continuous Primitive Integral

IfFis a continuous function on the real line andf=F′is its distributional derivative, then the continuous primitive integral of distributionfis∫abf=F(b)−F(a). This integral contains the Lebesgue, Henstock-Kurzweil, and wide Denjoy integrals. Under the Alexiewicz norm, the space of integrable distributions is a Banach space. We define the convolutionf∗g(x)=∫−∞∞f(x−y)g(y)dyforfan integrable distribution andga function of bounded variation or anL1function. Usual properties of convolutions are shown to hold: commutativity, associativity, commutation with translation. Forgof bounded variation,f∗gis uniformly continuous and we have the estimate‖f∗g‖∞≤‖f‖‖g‖ℬ𝒱, where‖f‖=supI|∫If|is the Alexiewicz norm. This supremum is taken over all intervalsI⊂ℝ. Wheng∈L1, the estimate is‖f∗g‖≤‖f‖‖g‖1. There are results on differentiation and integration of convolutions. A type of Fubini theorem is proved for the continuous primitive integral.

On the existence of Levi Foliations

Let L <img src="http:/img/fbpe/aabc/v73n1/0059c.gif"> <img src="http:/img/fbpe/aabc/v73n1/0059c2.gif"> be a real 3 dimensional analytic variety. For each regular point p <img src="http:/img/fbpe/aabc/v73n1/0059e.gif"> L there exists a unique complex line l p on the space tangent to L at p. When the field of complex line p <img ALIGN="MIDDLE" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img4.gif" ALT="$\displaystyle \mapsto$"> l p is completely integrable, we say that L is Levi variety. More generally; let L <img src="http:/img/fbpe/aabc/v73n1/0059c.gif"> M be a real subvariety in an holomorphic complex variety M. If there exists a real 2 dimensional integrable distribution on L which is invariant by the holomorphic structure J induced by M, we say that L is a Levi variety. We shall prove: Theorem. Let <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img5.gif" ALT="$ \cal {L}$"> be a Levi foliation and let <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img6.gif" ALT="$ \cal {F}$"> be the induced holomorphic foliation. Then, <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img6.gif" ALT="$ \cal {F}$"> admits a Liouvillian first integral. In other words, if <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img5.gif" ALT="$ \cal {L}$"> is a 3 dimensional analytic foliation such that the induced complex distribution defines an holomorphic foliation <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img6.gif" ALT="$ \cal {F}$">; that is, if <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img5.gif" ALT="$ \cal {L}$"> is a Levi foliation; then <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img6.gif" ALT="$ \cal {F}$"> admits a Liouvillian first integral--a function which can be constructed by the composition of rational functions, exponentiation, integration, and algebraic functions (Singer 1992). For example, if f is an holomorphic function and if theta is real a 1-form on <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img8.gif" ALT="$ \mathbb {R}$">; then the pull-back of theta by f defines a Levi foliation <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img5.gif" ALT="$ \cal {L}$"> : f*theta = 0 which is tangent to the holomorphic foliation <img ALIGN="BOTTOM" BORDER="0" src="http:/img/fbpe/aabc/v73n1/0059img6.gif" ALT="$ \cal {F}$"> : df = 0. This problem was proposed by D. Cerveau in a meeting (see Fernandez 1997).