Differential Geometry

In this paper we will discuss local coordinates canonically corresponding to a Kahler metric. We will also discuss and prove the C ∞ convergence of Bergman metrics following Tian's result on C 2 convergence of Bergman metrics. At the end, we present an interesting characterization of ample line bundle that could be useful in arithmetic geometry.

We show that there is a well-defined cap-product structure on the Fintushel-Stern spectral sequence. Hence we obtain the induced cap-product structure on the ${\BZ}_8$-graded instanton Floer homology. The cap-product structure provides an essentially new property of the instanton Floer homology, from a topological point of view, which multiplies a finite dimensional cohomology class by an infinite dimensional homology class (Floer cycles) to get another infinite dimensional homology class.

The group of real 4 by 4 upper triangular matrices with 1s on the diagonal has a left-invariant subRiemannian (or Carnot-Caratheodory) structure whose underlying distribution corresponds to the superdiagonal. We prove that the associated subRiemannian geodesic flow is not completely integrable. This provides the first example of a Carnot group (graded nilpotent Lie group with an invariant subRiemannian structure supported on the generating subspace) with a non-integrable geodesic flow. We apply this result to prove that the centralizer for the corresponding quadratic ``quantum'' Hamiltonian in the universal enveloping algebra for this group is ``as small as possible''.

There is a remarkable type of field of two-planes special to four dimensions known as an Engel distributions. They are the only stable regular distributions besides the contact, quasi-contact and line fields. If an arbitrary two-plane field on a four-manifold is slightly perturbed then it will be Engel at generic points. On the other hand, if a manifold admits an oriented Engel structure then the manifold must be parallelizable and consequently the alleged Engel distribution must have a degeneration loci -- a point set where the Engel conditions fails. By a theorem of Zhitomirskii this locus is a finite union of surfaces. We prove that these surfaces represent Chern classes associated to the distribution.

From the cohomological point of view the symplectomorphism group Sympl(M) of a symplectic manifold is `` tamer'' than the diffeomorphism group. The existence of invariant polynomials in the Lie algebra sympl(M) , the symplectic Chern-Weil theory, and the existence of Chern-Simons-type secondary classes are first manifestations of this principles. On a deeper level live characteristic classes of symplectic actions in periodic cohomology and symplectic Hodge decompositions. The present paper is called to introduce theories and constructions listed above and to suggest numerous concrete applications. These includes: nonvanishing results for cohomology of symplectomorphism groups (as a topological space, as a topological group and as a discrete group), symplectic rigidity of Chern classes, lower bounds for volumes of Lagrangian isotopies, the subject started by Givental, Kleiner and Oh, new characters for Torelli group and generalizations for automorphism groups of one-relator groups, arithmetic properties of special values of Witten zeta-function and solution of a conjecture of Brylinski. The Appendix, written by L. Katzarkov, deals with fixed point sets of finite group actions in moduli spaces.

Let X be a closed manifold with zero Euler characteristic, and let f: X --> S^1 be a circle-valued Morse function. We define an invariant I which counts closed orbits of the gradient of f, together with flow lines between the critical points. We show that our invariant equals a form of topological Reidemeister torsion defined by Turaev [Math. Res. Lett. 4 (1997) 679-695]. We proved a similar result in our previous paper [Topology, 38 (1999) 861-888], but the present paper refines this by separating closed orbits and flow lines according to their homology classes. (Previously we only considered their intersection numbers with a fixed level set.) The proof here is independent of the previous proof and also simpler. Aside from its Morse-theoretic interest, this work is motivated by the fact that when X is three-dimensional and b_1(X)>0, the invariant I equals a counting invariant I_3(X) which was conjectured in our previous paper to equal the Seiberg-Witten invariant of X. Our result, together with this conjecture, implies that the Seiberg-Witten invariant equals the Turaev torsion. This was conjectured by Turaev [Math. Res. Lett. 4 (1997) 679-695] and refines the theorem of Meng and Taubes [Math. Res. Lett. 3 (1996) 661-674].

Let X be a compact oriented Riemannian manifold and let ϕ:X→ S 1 be a circle-valued Morse function. Under some mild assumptions on ϕ , we prove a formula relating: (a) the number of closed orbits of the gradient flow of ϕ of any given degree; (b) the torsion of a ``Morse complex'', which counts gradient flow lines between critical points of ϕ ; and (c) a kind of Reidemeister torsion of X determined by the homotopy class of ϕ . When dim(X)=3 and b 1 (X)>0 , we state a conjecture analogous to Taubes's ``SW=Gromov'' theorem, and we use it to deduce (for closed manifolds, modulo signs) the Meng-Taubes relation between the Seiberg- Witten invariants and the ``Milnor torsion'' of X.

An example of mechanical system whose configuration space is direct product of a curved space and the local group of rotations, is presented. The system is considered as a model of spinning particle moving in the space. The Hamiltonian formalism for this system and possible method for its quantization are discussed. It is shown that the Hamilton equations coincide with the Papapetrou equations for spinning test-particle in general relativity.

We extend the calculus of multiplicative vector fields and differential forms and their intrinsic derivatives from Lie groups to Lie groupoids; this generalization turns out to include also the classical process of complete lifting from arbitrary manifolds to tangent and cotangent bundles. Using this calculus we give a new description of the Lie bialgebroid structure associated with a Poisson groupoid.

The existence of the theory of `twisted cotangent bundles' (symplectic groupoids) allows to study classical mechanical systems which are generalized in the sense that their configurations form a Poisson manifold. It is natural to study from this point of view first such systems which arise in the context of some basic physical symmetry (space-time, rotations, etc.). We review results obtained so far in this direction.