This paper relates the boundary term in the Chern-Gauss-Bonnet formula on 4-manifolds M with the renormalized volume V , as defined in the AdS/CFT correspondence, for asymptotically hyperbolic Einstein metrics on M . In addition we show that the differential or variation dV of V , or equivalently the variation of the L norm of the Weyl curvature, is intrinsically determined by the conformal inf...

In this paper, we give a new formulation of invariant theory for elliptic Weyl group using the group O(2, n). As an elliptic Weyl group quotient, we define a suitable C∗-bundle. We show that it has a conformal Frobenius structure which we define in this paper. Then its good section could be identified with a Frobenius manifold which we constructed in [13].

The standard eigenfunctions φλ = ei〈λ,x〉 on flat tori Rn/L have L-norms bounded independently of the eigenvalue. In the case of irrational flat tori, it follows that L2normalized eigenfunctions have uniformly bounded L-norms. Similar bases exist on other flat manifolds. Does this property characterize flat manifolds? We give an affirmative answer for compact Riemannian manifolds with quantum co...

It is proved that any compact almost Kähler, Einstein 4-manifold whose fundamental form is a root of the Weyl tensor is necessarily Kähler.

We study some aspects of noncommutative differential geometry on a finite Weyl group in the sense of S. Woronowicz, K. Bresser et al., and S. Majid. For any finite Weyl group W we consider the subalgebra generated by flat connections in the left-invariant exterior differential algebra of W. For root systems of type A and D we describe a set of relations between the flat connections, which conje...

In this article we verify an orbifold version of a conjecture of Nimershiem from 1998. Namely, for every flat n–manifold M, we show that the set of similarity classes of flat metrics on M which occur as a cusp cross-section of a hyperbolic (n+1)–orbifold is dense in the space of similarity classes of flat metrics on M. The set used for density is precisely the set of those classes which arise i...

Construction of an infinite dimensional differentiable manifold R∞ not modelled on any Banach space is proposed. Definition, metric and differential structures of a Weyl algebra (P ∗ p M [[~]], ◦) and a Weyl algebra bundle (P∗M[[~]], ◦) are presented. Continuity of the ◦-product in the Tichonov topology is proved. Construction of the ∗-product of the Fedosov type in terms of theory of connectio...

The G-function associated to the semisimple Frobenius manifold C/W (where W is a Coxeter group or an extended affine Weyl group) is studied. The general form of the Gfunction is given in terms of a logarithmic singularity over caustics in the manifold. The main result in this paper is a universal formula for the G-function corresponding to the Frobenius manifold C/W̃ (An−1) , where W̃ (An−1) is a...