Bianchi identities in higher dimensions

A higher dimensional frame formalism is developed in order to study implications of the Bianchi identities for the Weyl tensor in vacuum spacetimes of the algebraic types III and N in arbitrary dimension $n$. It follows that the principal null congruence is geodesic and expands isotropically in two dimensions and does not expand in $n-4$ spacelike dimensions or does not expand at all. It is shown that the existence of such principal geodesic null congruence in vacuum (together with an additional condition on twist) implies an algebraically special spacetime. We also use the Myers-Perry metric as an explicit example of a vacuum type D spacetime to show that principal geodesic null congruences in vacuum type D spacetimes do not share this property.Comment: 25 pages, v3: Corrections to Appendix B as given in Erratum-ibid.24:1691,2007 are now incorporated (A factor of 2 was missing in certain Bianchi equations.

The structure of non-spacelike geodesics in dust collapse

We study here the behaviour of non-spacelike geodesics in dust collapse models in order to understand the casual structure of the spacetime. The geodesic families coming out, when the singularity is naked, corresponding to different initial data are worked out and analyzed. We also bring out the similarity of the limiting behaviour for different types of geodesics in the limit of approach to the singularity.Comment: 23 pages, 6 figures, to appear in PR

Non-commutative Hilbert modular symbols

The main goal of this paper is to construct non-commutative Hilbert modular symbols. However, we also construct commutative Hilbert modular symbols. Both the commutative and the non-commutative Hilbert modular symbols are generalizations of Manin's classical and non-commutative modular symbols. We prove that many cases of (non-)commutative Hilbert modular symbols are periods in the sense on Kontsevich-Zagier. Hecke operators act naturally on them. Manin defines the non-commutative modilar symbol in terms of iterated path integrals. In order to define non-commutative Hilbert modular symbols, we use a generalization of iterated path integrals to higher dimensions, which we call iterated integrals on membranes. Manin examines similarities between non-commutative modular symbol and multiple zeta values both in terms of infinite series and in terms of iterated path integrals. Here we examine similarities in the formulas for non-commutative Hilbert modular symbol and multiple Dedekind zeta values, recently defined by the author, both in terms of infinite series and in terms of iterated integrals on membranes.Comment: 50 pages, 5 figures, substantial improvement of the article arXiv:math/0611955 [math.NT], the portions compared to the previous version are: Hecke operators, periods and some categorical construction

Combinatorial and topological phase structure of non-perturbative n-dimensional quantum gravity

We provide a non-perturbative geometrical characterization of the partition function of $n$-dimensional quantum gravity based on a coarse classification of riemannian geometries. We show that, under natural geometrical constraints, the theory admits a continuum limit with a non-trivial phase structure parametrized by the homotopy types of the class of manifolds considered. The results obtained qualitatively coincide, when specialized to dimension two, with those of two-dimensional quantum gravity models based on random triangulations of surfaces.Comment: 13 page

On Cannon cone types and vector-valued multiplicative functions for genus-two-surface-group

We consider Cannon cone types for a surface group of genus $g$, and we give algebraic criteria for establishing the cone type of a given cone and of all its sub-cones. We also re-prove that the number of cone types is exactly $8g(2g - 1)+1.$ In the genus $2$ case, we explicitly provide the $48\times 48$ matrix of cone types, $M,$ and we prove that $M$ is primitive, hence Perron-Frobenius. Finally we define vector-valued multiplicative functions and we show how to compute their values by means of $M$