Modified differentials and basic cohomology for Riemannian foliations

We define a new version of the exterior derivative on the basic forms of a Riemannian foliation to obtain a new form of basic cohomology that satisfies Poincar\'e duality in the transversally orientable case. We use this twisted basic cohomology to show relationships between curvature, tautness, and vanishing of the basic Euler characteristic and basic signature.Comment: 20 pages, references added, minor corrections mad

Tautness for riemannian foliations on non-compact manifolds

For a riemannian foliation $\mathcal{F}$ on a closed manifold $M$, it is known that $\mathcal{F}$ is taut (i.e. the leaves are minimal submanifolds) if and only if the (tautness) class defined by the mean curvature form $\kappa_\mu$ (relatively to a suitable riemannian metric $\mu$) is zero. In the transversally orientable case, tautness is equivalent to the non-vanishing of the top basic cohomology group $H^{^{n}}(M/\mathcal{F})$, where n = \codim \mathcal{F}. By the Poincar\'e Duality, this last condition is equivalent to the non-vanishing of the basic twisted cohomology group $H^{^{0}}_{_{\kappa_\mu}}(M/\mathcal{F})$, when $M$ is oriented. When $M$ is not compact, the tautness class is not even defined in general. In this work, we recover the previous study and results for a particular case of riemannian foliations on non compact manifolds: the regular part of a singular riemannian foliation on a compact manifold (CERF).Comment: 18 page

Loop Groups, Kaluza-Klein Reduction and M-Theory

We show that the data of a principal G-bundle over a principal circle bundle is equivalent to that of a \hat{LG} = U(1) |x LG bundle over the base of the circle bundle. We apply this to the Kaluza-Klein reduction of M-theory to IIA and show that certain generalized characteristic classes of the loop group bundle encode the Bianchi identities of the antisymmetric tensor fields of IIA supergravity. We further show that the low dimensional characteristic classes of the central extension of the loop group encode the Bianchi identities of massive IIA, thereby adding support to the conjectures of hep-th/0203218.Comment: 26 pages, LaTeX, utarticle.cls, v2:clarifications and refs adde

Group elastic symmetries common to continuum and discrete defective crystals

The Lie group structure of crystals which have uniform continuous distributions of dislocations allows one to construct associated discrete structures—these are discrete subgroups of the corresponding Lie group, just as the perfect lattices of crystallography are discrete subgroups of R 3 , with addition as group operation. We consider whether or not the symmetries of these discrete subgroups extend to symmetries of (particular) ambient Lie groups. It turns out that those symmetries which correspond to automorphisms of the discrete structures do extend to (continuous) symmetries of the ambient Lie group (just as the symmetries of a perfect lattice may be embedded in ‘homogeneous elastic’ deformations). Other types of symmetry must be regarded as ‘inelastic’. We show, following Kamber and Tondeur, that the corresponding continuous automorphisms preserve the Cartan torsion, and we characterize the discrete automorphisms by a commutativity condition, (6.14), that relates (via the matrix exponential) to the dislocation density tensor. This shows that periodicity properties of corresponding energy densities are determined by the dislocation density