zw Let G be a finite abelian group and 1 ≤ r < ∞. We prove that every locally definable CG manifold admits a unique locally definable C∞G manifold structure up to locally definable C∞G diffeomorphism. zh 2000 Mathematics Subject Classification. 14P10, 14P20, 57S05, 57S15, 58A05, 58A07, 03C64.

A Calabi-Yau manifold is a complex Kähler manifold with trivial canonical bundle. In the attempt to construct such manifolds it is useful to take into consideration singular CalabiYaus. One of the simplest singularities which can arise is an orbifold singularity. An orbifold is the quotient of a smooth Calabi-Yau manifold by a discrete group action which generically has fixed points. Locally su...

A G-equivariant spinc structure on a manifold gives rise to a virtual representation of the group G, called the spinc quantization of the manifold. We present a cutting construction for S-equivariant spinc manifolds, and show that the quantization of the original manifold is isomorphic to the direct sum of the quantizations of the cut spaces. Our proof uses Kostanttype formulas, which express t...

A strict quantization of a Poisson manifold P on a subset I ⊆ R containing 0 as an accumulation point is defined as a continuous field of C∗-algebras {Ah̄}h̄∈I , with A0 = C0(P ), a dense subalgebra Ã0 of C0(P ) on which the Poisson bracket is defined, and a set of continuous cross-sections {Q(f )} f∈Ã0 for which Q0(f ) = f . Here Qh̄(f ∗) = Qh̄(f )∗ for all h̄ ∈ I , whereas for h̄ → 0 one requires t...

Let (M, f,G) be a manifold, a function and a Riemann metric on the manifold. Topologists would use these data in order to analyze the manifold by means of Morse theory, that is by studying the dynamical system ẋ = ±∇f . Many recent applications of physics to topology are based on another point of view suggested in E. Witten’s paper Supersymmetry and Morse theory J. Diff. Geom. (1982). Given the...

in the first part of this paper, some theorems are given for a riemannian manifold with semi-symmetric metric connection. in the second part of it, some special vector fields, for example, torse-forming vector fields, recurrent vector fields and concurrent vector fields are examined in this manifold. we obtain some properties of this manifold having the vectors mentioned above.

This is a survey of recent contributions to the area of special Kähler geometry. It is based on lectures given at the 21st Winter School on Geometry and Physics held in Srni in January 2001. 1 Remarkable features of special Kähler manifolds A (pseudo-) Kähler manifold (M,J, g) is a differentiable manifold endowed with a complex structure J and a (pseudo-) Riemannian metric g such that (i) J is ...

A symplectic groupoid G. := (G1 ⇉ G0) determines a Poisson structure on G0. In this case, we call G. a symplectic groupoid of the Poisson manifold G0. However, not every Poisson manifold M has such a symplectic groupoid. This keeps us away from some desirable goals: for example, establishing Morita equivalence in the category of all Poisson manifolds. In this paper, we construct symplectic Wein...