title: Stiefel-Whitney Currents

abstract:
  A canonically defined mod 2 linear dependency current is associated to each
collection of m sections of a real rank n vector bundle. This current is
supported on the linear dependency set of the collection of sections. It is
defined whenever the collection satisfies a weak measure theoretic condition
called "atomicity". Essentially any reasonable collection of sections satisfies
this condition, vastly extending the usual general position hypothesis. This
current is a mod 2, closed, locally integrally flat current of degree q= n-m+1
and hence determines a mod 2 cohomology class. This class is shown to be well
defined independent of the collection of sections. Moreover it is the q-th
Stielfel-Whitney class of the vector bundle. More is true if q is odd or q=n.
In this case a linear dependency current which is twisted by the orientation of
the bundle can be associated to the collection. The cohomology class of the
linear dependency current is 2-torsion and is the q-th twisted integral
Stiefel-Whitney class of the bundle. In addition, higher dependency and general
degeneracy currents of bundle maps are studied, together with applications to
singularities of projections and maps. These results rely on a theorem of
Federer which states that the complex of integrally flat currents mod p
computes cohomology mod p. An alternate approach to Federer's theorem is
offered in an appendix. This approach is simpler and is via sheaf theory.


title: On constant mean curvature surfaces with periodic metric

abstract:
  We investigate CMC-surfaces with periodic metric in a dressing orbit of the
cylinder. It is shown, that such surfaces are always of finite type. Using the
periodicity conditions for the extended frame of a CMC-surface, we develop an
alternative approach to the classification of CMC-tori given by Pinkall and
Sterling.


title: Great sphere foliations and manifolds with curvature bounded above

abstract:
  The survey is devoted to Toponogov's conjecture, that {\it if a complete
simply connected Riemannian manifold with sectional curvature $\le 4$ and
injectivity radius $\ge \pi/2$ has extremal diameter $\pi/2$, then it is
isometric to CROSS}. In Section 1 the relations of problem with geodesic
foliations of a round sphere are considered, but the proof of conjecture on
this way is not complete. In Section 2 the proof based on recent results and
methods for topology and volume of Blaschke manifolds is given.


title: Deforming a map into a harmonic map

abstract:
  Using a flow first introduced by J.P. Anderson, we obtain some existence
theorems for harmonic maps from a noncompact complete Riemannian manifold into
a complete Riemannian manifold. In particular, we prove as a corollary a recent
result of Hardt and Wolf stating that any quasisymmetric map of the sphere that
is sufficiently close to the identity can be extended to a quasiconformal
harmonic diffeomorphism of the hyperbolic ball.
  This version contains a much simpler proof than the first version.


title: On index formulas for manifolds with metric horns

abstract:
  In this paper we discuss the index problem for geometric differential
operators (Spin-Dirac operator, Gau{\ss}-Bonnet operator, Signature operator)
on manifolds with metric horns. On singular manifolds these operators in
general do not have unique closed extensions. But there always exist two
extremal extensions $D_{min}$ and $D_{max}$. We describe the quotient ${\cal
D}(D_{max}) / {\cal D}(D_{min})$ explicitely in geometric resp. topologic terms
of the base manifolds of the metric horns. We derive index formulas for the
Spin-Dirac and Gau{\ss}-Bonnet operator. For the Signature operator we present
a partial result.
  The first version of this paper was completed August 1995 at the University
of Augsburg.


title: On a theorem by do Carmo and Dajczer

abstract:
  We give a new proof of a theorem by M.P. do Carmo and M. Dajczer on
helicoidal surfaces of constant mean curvature.


title: The spectrum of Kleinian manifolds

abstract:
  A Kleinian manifold Y is a quotient of a rank-one symmetric space of
non-compact type by a convex-cocompact discrete group of isometries. We
describe the spectral decomposition of the space of square integrable sections
of locally homogeneous bundles on Y with respect to locally invariant
differential operators. In the course of the proof we obtain meromorphic
continuations of Eisenstein series and scattering matrices.


title: Quantization of symplectic tori in a real polarization

abstract:
  We apply the geometric quantization method with real polarizations to the
quantization of a symplectic torus. By quantizing with half-densities we
canonically associate to the symplectic torus a projective Hilbert space and
prove that the projective factor is expressible in terms of the
Maslov-Kashiwara index. As in the quantization of a linear symplectic space, we
have two ways of resolving the projective ambiguity: (i) by introducing a
metaplectic structure and using half-forms in the definition of the Hilbert
space; (ii) by choosing a 4-fold cover of the Lagrangian Grassmannian of the
linear symplectic space covering the torus. We show that the Hilbert space
constructed through either of these approaches realizes a unitary
representation of the integer metaplectic group.


title: Abelian Chern-Simons theory

abstract:
  We give a construction of the abelian Chern-Simons gauge theory from the
point of view of a 2+1 dimensional topological quantum field theory. The
definition of the quantum theory relies on geometric quantization ideas which
have been previously explored in connection to the nonabelian Chern-Simons
theory [JW,ADW]. We formulate the topological quantum field theory in terms of
the category of extended 2- and 3-manifolds introduced by Walker [Wa] and prove
that it satisfies the axioms of unitary topological quantum field theories
formulated by Atiyah [A1].


title: Determinant lines, von Neumann algebras and $L^2$ torsion

abstract:
  In this paper, we suggest a construction of determinant lines of finitely
generated Hilbertian modules over finite von Neumann algebras. Nonzero elements
of the determinant lines can be viewed as volume forms on the Hilbertian
modules. Using this, we study both $L^2$ combinatorial and $L^2$ analytic
torsion invariants associated to flat Hilbertian bundles over compact polyhedra
and manifolds; we view them as volume forms on the reduced $L^2$ homology and
cohomology. These torsion invariants specialize to the the classical
Reidemeister-Franz torsion and the Ray-Singer torsion in the finite dimensional
case. Under the assumption that the $L^2$ homology vanishes, the determinant
line can be canonically identified with $\R$, and our $L^2$ torsion invariants
specialize to the $L^2$ torsion invariants previously constructed by A.Carey,
V.Mathai and J.Lott. We also show that a recent theorem of Burghelea et al. can
be reformulated as stating equality between two volume forms (the combinatorial
and the analytic) on the reduced $L^2$ cohomology.


title: Groups quasi-isometric to symmetric spaces

abstract:
  We determine the structure of finitely generated groups which are
quasi-isometric to symmetric spaces of noncompact type, allowing Euclidean de
Rham factors If $X$ is a symmetric space of noncompact type with no Euclidean
de Rham factor, and $\Ga$ is a finitely generated group quasi-isometric to the
product $\E^k\times X$, then there is an exact sequence $1\ra H\ra\Ga\ra L\ra
1$ where $H$ contains a finite index copy of $\Z^k$ and $L$ is a uniform
lattice in the isometry group of $X$.


title:
Moduli Spaces of Embedded Constant Mean Curvature Surfaces with Few Ends
  and Special Symmetry

abstract:
  We give necessary conditions on complete embedded \cmc surfaces with three or
four ends subject to reflection symmetries. The respective submoduli spaces are
two-dimensional varieties in the moduli spaces of general \cmc surfaces. We
characterize fundamental domains of our \cmc surfaces by associated great
circle polygons in the three-sphere.


title: The Spinor Representation of Surfaces in Space

abstract:
  The spinor representation is developed for conformal immersions of Riemann
surfaces into space. We adapt the approach of Dennis Sullivan, which treats a
spin structure on a Riemann surface M as a complex line bundle S whose square
is the canonical line bundle K=T(M). Given a conformal immersion of M into
\bbR^3, the unique spin strucure on S^2 pulls back via the Gauss map to a spin
structure S on M, and gives rise to a pair of smooth sections (s_1,s_2) of S.
Conversely, any pair of sections of S generates a (possibly periodic) conformal
immersion of M under a suitable integrability condition, which for a minimal
surface is simply that the spinor sections are meromorphic. A spin structure S
also determines (and is determined by) the regular homotopy class of the
immersion by way of a \bbZ_2-quadratic form q_S. We present an analytic
expression for the Arf invariant of q_S, which decides whether or not the
correponding immersion can be deformed to an embedding. The Arf invariant also
turns out to be an obstruction, for example, to the existence of certain
complete minimal immersions. The later parts of this paper use the spinor
representation to investigate minimal surfaces with embedded planar ends. In
general, we show for a spin structure S on a compact Riemann surface M with
punctures at P that the space of all such (possibly periodic) minimal
immersions of M\setminus P into \bbR^3 (upto homothety) is the the product of
S^1\times H^3 with the Grassmanian of 2-planes in a complex vector space \calK
of meromorphic sections of S. An important tool -- a skew-symmetric form \Omega
defined by residues of a certain meromorphic quadratic differential on M --
lets us compute how \calK varies as M and P are varied. Then we apply this to
determine the moduli spaces of planar-ended minimal spheres and real projective
planes, and also to construct a new family of minimal tori and a minimal Klein
bottle with 4 ends. These surfaces compactify in S^3 to yield surfaces critical
for the \Moebius invariant squared mean curvature functional W. On the other
hand, Robert Bryant has shown all W-critical spheres and real projective planes
arise this way. Thus we find at the same time the moduli spaces of W-critical
spheres and real projective planes via the spinor representation.


title: Bonnet pairs and isothermic surfaces

abstract:
  In this note we classify all Bonnet pairs on a simply connected domain. Our
main intent was to apply what we call a quaternionic function theory to a
concrete problem in differential geometry. The ideas are simple: conformal
immersions into quaternions or imaginary quaternions take the place of chart
maps for a Riemann surface. Starting from a reference immersion we construct
all conformal immersions of a given (simply connected) Riemann surface (up to
translational periods) by spin transformations. With this viewpoint in mind we
discuss how to construct all Bonnet pairs on a simply connected domain from
isothermic surfaces and vice versa. Isothermic surfaces are solutions to a
certain soliton equation and thus a simple dimension count tells us that we
obtain Bonnet pairs which are not part of any of the classical Bonnet families.
The corresponcence between Bonnet pairs and isothermic surfaces is explicit and
to each isothermic surface we obtain a 4-parameter family of Bonnet pairs.


title: De Rham theorem for extended L^2-cohomology

abstract:
  We prove an analogue of the de Rham theorem for the extended L^2-cohomology
introduced by M. Farber. This is done by establishing that the de Rham complex
over a compact closed manifold with coefficients in a flat Hilbert bundle E of
A-modules over a finite von Neumann algebra A is chain-homotopy equivalent
(with bounded morphisms and homotopy operators) to a combinatorial complex with
the same coefficients. This is established by using the Witten deformation of
the de Rham complex. We also prove that the de Rham complex is chain-homotopy
equivalent to the spectrally truncated de Rham complex which is also finitely
generated.


title:
Transverse measures, the modular class, and a cohomology pairing for Lie
  algebroids

abstract:
  We show that every Lie algebroid $A$ over a manifold $P$ has a natural
representation on the line bundle $Q_A = \wedge^{top}A \otimes \wedge^{top}
T^*P$. The line bundle $Q_A$ may be viewed as the Lie algebroid analog of the
orientation bundle in topology, and sections of $Q_A$ may be viewed as
transverse measures to $A$. As a consequence, there is a well-defined class in
the first Lie algebroid cohomology $H^1(A)$ called the modular class of the Lie
algebroid $A$. This is the same as the one introduced earlier by Weinstein
using the Poisson structure on $A^*$. We show that there is a natural pairing
between the Lie algebroid cohomology spaces of $A$ with trivial coefficients
and with coefficients in $Q_A$. This generalizes the pairing used in the
Poincare duality of finite-dimensional Lie algebra cohomology. The case of
holomorphic Lie algebroids is also discussed, where the existence of the
modular class is connected with the Chern class of the line bundle $Q_A$.


title:
Coordinates on Schubert cells, Kostant's harmonic forms, and the
  Bruhat-Poisson structure on $G/B$

abstract:
  We relate Kostant's theorem on the cohomology of a flag manifold $G/B$ with
the geometry of the Bruhat-Poisson structure. We express Kostant's harmonic
forms in terms of the moment maps (for the torus action) and the Liouville
volume forms for the symplectic structures on the Schubert cells induced by the
Bruhat-Poisson structure. We do this by writing everything down explicitly in
some coordinates on each cell.


title: The Penrose transform on conformally Bochner-K\"ahler manifolds

abstract:
  We give a generalization of the Penrose transform on Hermitian manifolds with
metrics locally conformally equivalent to Bochner-K\"ahler metrics. We also
give an explicit formula for the inverse transform. This paper is a
generalization of "The Twistor correspondence of the Dolbeault complex over
$\C^n$" (dg-ga/9501004) and intended to supersede it.


title: Canonical coordinates and Bergman metrics

abstract:
  In this paper we will discuss local coordinates canonically corresponding to
a Kahler metric. We will also discuss and prove the $C^\infty$ 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.


title: Integrable geodesic flows of non-holonomic metrics

abstract:
  Normal geodesic flows flows of Carnot-Caratheodory are discussed from the
point of view of the theory of Hamiltonian systems. The geodesic flows
corresponding to left-invariant metrics and left- and -right-invariant rank 2
distributions on the three-dimensional Heisenberg group are analysed as
integrable systems. The flows corresponding to left-invariant metrics and
left-invariant distributions on Lie groups are reduced to Euler equations on
Lie groups. Relation of these constructions to problems of analytical mechanics
is discussed.


title: Surfaces of revolution in terms of solitons

abstract:
  The detailed analysis of the generalised Weierstrass representation of
surfaces of revolution and their deformations induced by the modified
Korteweg--de Vries (mKdV) equations is done. In particular, it is shown that
these deformations preserve tori. The geometric meaning of the potential of
surface is discussed and the deformations of potentials induced by geometric
deformations (the Moebius inversion and a passing to a dual surface) are
described. The conjecture of invariance of first integrals of the mKdV
hierarchy, considering as invariants of surfaces, with respect to confomal
changes of the metric of the ambient space is introduced. Some numeric
computations confirming it are also presented. This paper is the sequel to the
previous author's paper "Modified Novikov-Veselov equation and differential
geometry of surfaces" (dg-ga/9511005) where generic surfaces are considered.
Here the surfaces of revolution are regarded as a toy model for the general
case.


title:
Uniqueness of Positive Solutions of the Conformal Scalar Curvature
  Equation and Applications to Conformal Transformations

abstract:
  We study uniqueness of positive solutions to the conformal scalar curvature
equation on complete Riemannian manifolds with constant negative scalar
curvature. We apply the results to show that conformal transformations on
certain complete Riemannian manifolds of constant negative scalar curvature are
isometries. We also study uniqueness of complete positive solutions and radial
solutions.


title: Equivariant Novikov inequalities

abstract:
  We establish an equivariant generalization of the Novikov inequalities which
allow to estimate the topology of the set of critical points of a closed basic
invariant form by means of twisted equivariant cohomology of the manifold. We
apply these inequalities to study cohomology of the fixed points set of a
symplectic torus action. We show that in this case our inequalities are
perfect, i.e. they are in fact equalities.


title: Von Neumann categories and extended $L^2$ cohomology

abstract:
  In this paper we suggest a new general formalism for studying the invariants
of polyhedra and manifolds comming from the theory of von Neumann algebras.
First, we examine generality in which one may apply the construction of the
extended abelian category, which was suggested in the previous publications of
the author, using the ideas of P.Freyd. This leads to the notions of a finite
von Neumann category and of a trace on such category. Given a finite von
Neumann category, we study the extended homology and cohomology theories with
values in the abelian extension. Any trace on the initial category produces
numerical invariants - the von Neumann dimension and the Novikov - Shubin
numbers. Thus, we obtain the local versions of the Novikov - Shubin invariants,
localized at different traces. In the "abelian" case this localization can be
made more geometric: we show that any torsion object determines a "divisor" --
a closed subspace of the space of the parameters. The divisors of torsion
objects together with the information produced by the local Novikov - Shubin
invariants may be used to study multiplicities of intersections of algebraic
and analytic varieties (we discuss here only simple examples demonstrating this
possibility). We compute explicitly the divisors and the von Neumann dimensions
of the extended cohomology in the real analytic situation. We also give general
formulae for the extended cohomology of a mapping torus. Finally, we show how
one can define a De Rham version of the extended cohomology and prove a De Rham
type theorem.


title: Twistor Bundles, Einstein Equations and Real Structures

abstract:
  We consider sphere bundles P and P' of totally null planes of maximal
dimension and opposite self-duality over a 4-dimensional manifold equipped with
a Weyl or Riemannian geometry. The fibre product PP' of P and P' is found to be
appropriate for the encoding of both the selfdual and the Einstein-Weyl
equations for the 4-metric. This encoding is realized in terms of the
properties of certain well defined geometrical objects on PP'. The formulation
is suitable for both complex- and real-valued metrics. It unifies results for
all three possible real signatures. In the purely Riemannian positive definite
case it implies the existence of a natural almost hermitian structure on PP'
whose integrability conditions correspond to the self-dual Einstein equations
of the 4-metric. All Einstein equations for the 4-metric are also encoded in
the properties of this almost hermitian structure on PP'.


title: Twisted L^2 invariants of non-simply connected manifolds

abstract:
  We develop the theory of twisted L^2-cohomology and twisted spectral
invariants for flat Hilbertian bundles over compact manifolds. They can be
viewed as functions on the first de Rham cohomology of M and they generalize
the standard notions. A new feature of the twisted L^2-cohomology theory is
that in addition to satisfying the standard L^2 Morse inequalities, they also
satisfy certain asymptotic L^2 Morse inequalities. These reduce to the standard
Morse inequalities in the finite dimensional case, and when the Morse 1-form is
exact. We define the extended twisted L^2 de Rham cohomology and prove the
asymptotic L^2 Morse-Farber inequalities, which give quantitative lower bounds
for the Morse numbers of a Morse 1-form on M.


title: Von Neumann spectra near the spectral gap

abstract:
  In this paper we study some new von Neumann spectral invariants associated to
the Laplacian acting on L^2 differential forms on the universal cover of a
closed manifold. These invariants coincide with the Novikov-Shubin invariants
whenever there is no spectral gap in the spectrum of the Laplacian, and are
homotopy invariants in this case. In the presence of a spectral gap, they
differ in character and value from the Novikov-Shubin invariants. Under a
positivity assumption on these invariants, we prove that certain L^2 theta and
L^2 zeta functions defined by metric dependent combinatorial Laplacians acting
on $L^2$ cochains associated with a triangulation of the manifold, converge
uniformly to their analytic counterparts, as the mesh of the triangulation goes
to zero.


title:
Asymptotic properties of energy of harmonic maps on asymptotically
  hyperbolic manifolds

abstract:
  Asymptotic behavior of energy of a harmonic map defined on an asymptotically
hyperbolic manifold is considered. Using the growth of energy, we show that a
harmonic map defined on some asymptotically hyperbolic manifolds has to be
constant if the total energy is finite, or if the map approaches a point fast
enough, in terms of a defining function for the boundary.


title: Dirac structures and Poisson homogeneous spaces

abstract:
  Poisson homogeneous spaces for Poisson groupoids are classfied in terms of
Dirac structures for the corresponding Lie bialgebroids. Applications include
Drinfel'd's classification in the case of Poisson groups and a description of
leaf spaces of foliations as homogeneous spaces of pair groupoids.


title:
Equivariant Poisson Cohomology and a Spectral Sequence Associated with a
  Moment Map

abstract:
  We introduce and study a new spectral sequence associated with a Poisson
group action on a Poisson manifold and an equivariant momentum mapping. This
spectral sequence is a Poisson analog of the Leray spectral sequence of a
fibration. The spectral sequence converges to the Poisson cohomology of the
manifold and has the $E_2$-term equal to the tensor product of the cohomology
of the Lie algebra and the equivariant Poisson cohomology of the manifold. The
latter is defined as the equivariant cohomology of the multi-vector fields made
into a G-differential complex by means of the momentum mapping. An extensive
introduction to equivariant cohomology of G-differential complexes is given
including some new results and a number of examples and applications are
considered.


title: Pseudo-orbits, pseudoleaves and geometric entropy of foliations

abstract:
  We show that the entropy of a finitely generated pseudogroup (resp., of a
foliation of a compact Riemannian manifold) can be calculated by suitable
counting separated pseudo-orbits (resp., pseudoleaves).


title:
Nijenhuis tensors and obstructions for pseudoholomorphic mapping
  constructions

abstract:
  In this paper we present some approaches to classification of almost complex
structures and to construction of local or formal pseudoholomorphic mapping
from one almost complex manifold to another. The corresponding criteria are
given in terms of Nijenhuis tensors and their generalizations.
  We deal with the prolongations of the Cauchy-Riemann equation in 1-jets of
the mapping from one almost complex manifold to another. We give a criterion of
the prolongation existence of k-th pseudoholomorphic jets to the (k+1)-th ones.
As a consequence basing on the Kuranishi's approach we obtain formal normal
forms of almost complex structures. As another consequence we get the formal
part of Newlander-Nirenberg's theorem on integrability.
  We also introduce the notion and study the properties of the linear Nijenhuis
tensor, which we apply to almost complex structures of general position. Two
applications are exhibited: a classification of four-dimensional almost complex
structures in terms of distributions and a characterization of the new class of
structures which we call Lie almost complex structures.


title:
Some classificational problems in four-dimensional geometry:
  distributions, almost complex structures and Monge-Ampere equations

abstract:
  In this paper we consider three deeply connected classificational problems on
four-dimensional manifolds. First we consider and describe locally regular
distributions. Second we give a classification of almost complex structures of
general position in terms of distributions. Finally we classify nondegenerate
Monge-Ampere equations with two variables in terms of e-structures.


title:
Projectively invariant symbol map and cohomology of vector fields Lie
  algebras intervening in quantization

abstract:
  We define the unique (up to normalization) symbol map from the space of
linear differential operators on $R^n$ to the space of polynomial on fibers
functions on $T^* R^n$, equivariant with respect to the Lie algebra of
projective transformations $sl_{n+1}\subset\Vect(R^n)$. We apply the
constructed $sl_{n+1}$-invariant symbol to studying of the natural
one-parameter family of $\Vect(M)$-modules on the space of linear differential
operators on an arbitrary manifold M. Each of the $\Vect(M)$-action from this
family can be interpreted as a deformation of the standard $\Vect(M)$-module
$S(M)$ of symmetric contravariant tensor fields on M. We define (and
calculatein the case: $M= R^n$) the corresponding cohomology of $\Vect(M)$
related with this deformation. This cohomology realize the obstruction for
existence of equivariant symbol and quantization maps. The projective Lie
algebra $sl_{n+1}$ naturally appears as the algebra of symmetries on which the
involved $\Vect(M)$-cohomology is trivial.


title: On Young hulls of convex curves in R^{2n}

abstract:
  For a convex curve in an even-dimensional affine space we introduce a series
of convex domains (called Young hulls), describe their structure and give a
formulas fo the volume of the biggest of these domains. This paper is an
attempt to generalize the classical isoperimetric inequality for the volume of
the convex hull of a space curve due to I.Schoenberg.


title: Instantons and the information metric

abstract:
  The information metric arises in statistics as a natural inner product on a
space of probability distributions. In general this inner product is positive
semi-definite but is potentially degenerate.
  By associating to an instanton its energy density, we can examine the
information metric {\bf g} on the moduli spaces $\M$ of self-dual connections
over Riemannian 4-manifolds. Compared with the more widely known $L^2$ metric,
the information metric better reflects the conformal invariance of the
self-dual Yang-Mills equations, and seems to have better completeness
properties. In the case of $SU(2)$ instantons on $S^4$ of charge one, {\bf g}
is known to be the hyperbolic metric on the five-ball. We show more generally
that for charge-one $SU(2)$ instantons over $1$-connected, positive-definite
manifolds, {\bf g} is nondegenerate and complete in the collar region of $\M$,
and is `asymptotically hyperbolic' there; {\bf g} vanishes at the cone points
of $\M$. We give explicit formulae for the metric on the space of instantons of
charge one on $\C P_2$.


title: A discrete version of the Darboux transform for isothermic surfaces

abstract:
  We study Christoffel and Darboux transforms of discrete isothermic nets in
4-dimensional Euclidean space: definitions and basic properties are derived.
Analogies with the smooth case are discussed and a definition for discrete
Ribaucour congruences is given. Surfaces of constant mean curvature are special
among all isothermic surfaces: they can be characterized by the fact that their
parallel constant mean curvature surfaces are Christoffel and Darboux
transforms at the same time. This characterization is used to define discrete
nets of constant mean curvature. Basic properties of discrete nets of constant
mean curvature are derived.


title: Projectively flat Finsler 2-spheres of constant curvature

abstract:
  I classify the Finsler structures on the 2-sphere that have constant
Finsler-Gauss curvature and whose geodesics are the great circles. Modulo
diffeomorphism, there is a 2-parameter family of such Finsler structures, only
one of which is homogeneous or symmetric, namely the Riemannian one.
  I discuss the history of the problem and its relation with Hilbert's Fourth
Problem and the calculus of variations.


title: The spherical phylon group and invariants of the Laplace transform

abstract:
  We introduce the spherical phylon group, a subgroup of the group of all
formal diffeomorphisms of $\R^d$ that fix the origin. The invariant theory of
the spherical phylon group is used to understand the invariants of the Laplace
transform.


title: Heat kernel and moduli spaces II

abstract:
  In this paper we continue our study on the moduli spaces of flat G-bundles,
for any semi-simple Lie group G, over a Riemann surface by using heat kernel
and Reidemeister torsion. Formulas for intersection numbers on the moduli
spaces over a Riemann surface with several boundary components, over
non-orientable Riemann surfaces are obtained. Some general vanishing theorems
about characteristic numbers of the moduli spaces are proved. We also extend
our method to study Higgs moduli spaces, to introduce invariants for knots and
3-manifolds.

categories: dg-ga alg-geom math.AG math.DG math.QA q-alg


title:
On quadratic and nonquadratic forms: Application to nonbijective R^{2m}
  -> R^{2m-n} transformations

abstract:
  Hurwitz transformations are defined as specific automorphisms of a
Cayley-Dickson algebra. These transformations generate quadratic and
nonquadratic forms. We investigate here the Hurwitz transformations
corresponding to Cayley-Dickson algebras of dimensions 2m = 2, 4 and 8. The
Hurwitz transformations which lead to quadratic forms are discussed from
geometrical and Lie-algebraic points of view. Applications to number theory and
dynamical systems are briefly examined.


title: Delocalized $L^2$-Invariants

abstract:
  We define extensions of the $L^2$-analytic invariants of closed manifolds,
called delocalized $L^2$-invariants. These delocalized invariants are
constructed in terms of a nontrivial conjugacy class of the fundamental group.
We show that in many cases, they are topological in nature. We show that the
marked length spectrum of an odd-dimensional hyperbolic manifold can be
recovered from its delocalized $L^2$-analytic torsion. There are technical
convergence questions.


title:
Circle-valued Morse theory, Reidemeister torsion, and Seiberg-Witten
  invariants of 3-manifolds

abstract:
  Let X be a compact oriented Riemannian manifold and let $\phi:X\to S^1$ be a
circle-valued Morse function. Under some mild assumptions on $\phi$, we prove a
formula relating:
  (a) the number of closed orbits of the gradient flow of $\phi$ of any given
degree;
  (b) the torsion of a ``Morse complex'', which counts gradient flow lines
between critical points of $\phi$; and
  (c) a kind of Reidemeister torsion of X determined by the homotopy class of
$\phi$.
  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.


title: Classical mechanical systems based on Poisson symmetry

abstract:
  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.


title: Free motion on the Poisson plane and sphere

abstract:
  Poisson plane and sphere --- homogeneous spaces of Poisson groups E(2) and
SU(2) (resp.) --- have phase spaces (corresponding symplectic groupoids), in
which a free Hamiltonian is naturally defined. We solve the equations of motion
and point out some unexpected features: free motion on the plane is bounded
(periodic) and free trajectories on the sphere are all circles except the big
ones.


title: Twisted Legendre transformation

abstract:
  The general framework of Legendre transformation is extended to the case of
symplectic groupoids, using an appropriate generalization of the notion of
generating function (of a Lagrangian submanifold).


title: Free motion on the Poisson SU(n) group

abstract:
  SL(N,C) is the phase space of the Poisson SU(N). We calculate explicitly the
symplectic structure of SL(N,C), define an analogue of the Hamiltonian of the
free motion on SU(N) and solve the corresponding equations of motion. Velocity
is related to the momentum by a non-linear Legendre transformation.


title:
Almost Complex and Almost Product Einstein Manifolds from a Variational
  Principle

abstract:
  It is shown that the first order (Palatini) variational principle for a
generic nonlinear metric-affine Lagrangian depending on the (symmetrized) Ricci
square invariant leads to an almost-product Einstein structure or to an
almost-complex anti-Hermitian Einstein structure on a manifold. It is proved
that a real anti-Hermitian metric on a complex manifold satisfies the K\"ahler
condition on the same manifold treated as a real manifold if and only if the
metric is the real part of a holomorphic metric. A characterisation of
anti-K\"ahler Einstein manifolds and almost-product Einstein manifolds is
obtained. Examples of such manifolds are considered.


title: Long time behavior of leafwise heat flow for Riemannian foliations

abstract:
  For any Riemannian foliation F on a closed manifold M with an arbitrary
bundle-like metric, leafwise heat flow of differential forms is proved to
preserve smoothness on M at infinite time. This result and its proof have
consequences about the space of bundle-like metrics on M, about the dimension
of the space of leafwise harmonic forms, and mainly about the second term of
the differentiable spectral sequence of F.


title: The normal symbol on Riemannian manifolds

abstract:
  For an arbitrary Riemannian manifold $X$ and Hermitian vector bundles $E$ and
$F$ over $X$ we define the notion of the normal symbol of a pseudodifferential
operator $P$ from $E$ to $F$. The normal symbol of $P$ is a certain smooth
function from the cotangent bundle $T^*X$ to the homomorphism bundle $Hom
(E,F)$ and depends on the metric structures resp. the corresponding connections
on $X$, $E$ and $F$. It is shown that by a natural integral formula the
pseudodifferential operator $P$ can be recovered from its symbol. Thus, modulo
smoothing operators resp. smoothing symbols, we receive a linear bijective
correspondence between the space of symbols and the space of pseudodifferential
operators on $X$. This correspondence comprises a natural transformation
between appropriate functors. A formula for the asymptotic expansion of the
product symbol of two pseudodifferential operators in terms of the symbols of
its factors is given. Furthermore an expression for the symbol of the adjoint
is derived. Finally the question of invertibility of pseudodifferential
operators is considered. For that we use the normal symbol to establish a new
and general notion of elliptic pseudodifferential operators on manifolds.


title: Morse theory on the loop space of flat tori and symplectic Floer theory

abstract:
  We use closed geodesics to construct and compute Bott-type Morse homology
groups for the energy functional on the loop space of flat $n$-dimensional
tori, $n\ge 1$, and Bott-type Floer cohomology groups for their cotangent
bundles equipped with the natural symplectic structure. Both objects are
isomorpic to the singular homology of the loop space. In an appendix we perturb
the equations in order to eliminate degeneracies and to get to a situation with
nondegenerate critical points only. The (co)homology groups turn out to be
invariant under the perturbation.


title:
Harmonic morphisms from three-dimensional Euclidean and spherical space
  forms

abstract:
  P. Baird and the second author studied harmonic morphisms from a
three-dimensional simply-connected space form to a surface and obtained a
complete local and global classification of them. In this paper, we obtain a
description of all harmonic morphisms from any three-dimensional Euclidean and
spherical space form to a surface, namely that any such harmonic morphism is
the composition of a standard harmonic morphism and a weakly conformal map. We
list the standard harmonic morphisms.


title: Knots, Links, and 4-Manifolds

abstract:
  In this paper we investigate the relationship between isotopy classes of
knots and links in S^3 and the diffeomorphism types of homeomorphic smooth
4-manifolds. As a corollary of this initial investigation, we begin to uncover
the surprisingly rich structure of diffeomorphism types of manifolds
homeomorphic to the K3 surface.


title: Asimptotic dimension and Novikov-Shubin invariants for open manifolds

abstract:
  A trace on the C^*-algebra A of quasi-local operators on an open manifold is
described, based on the results in \cite{RoeOpen}. It allows a description `a
la Novikov-Shubin \cite{NS2} of the low frequency behavior of the
Laplace-Beltrami operator. The 0-th Novikov-Shubin invariant defined in terms
of such a trace is proved to coincide with a metric invariant, which we call
asymptotic dimension, thus giving a large scale ``Weyl asymptotics'' relation.
Moreover, in analogy with the Connes-Wodzicki result \cite{CoCMP,Co,Wo}, the
asymptotic dimension d measures the singular traceability (at 0) of the
Laplace-Beltrami operator, namely we may construct a (type II_1) singular trace
which is finite on the $^*$-bimodule over A generated by $\Delta^{-d/2}$.


title:
Intersection numbers on moduli spaces and symmetries of a Verlinde
  formula

abstract:
  We investigate the geometry and topology of a standard moduli space of stable
bundles on a Riemann surface, and use a generalization of the Verlinde formula
to derive results on intersection pairings.


title: SU(n)-Connections and Noncommutative Differential Geometry

abstract:
  We study the noncommutative differential geometry of the algebra of
endomorphisms of any SU(n)-vector bundle. We show that ordinary connections on
such SU(n)-vector bundle can be interpreted in a natural way as a
noncommutative 1-form on this algebra for the differential calculus based on
derivations. We interpret the Lie algebra of derivations of the algebra of
endomorphisms as a Lie algebroid. Then we look at noncommutative connections as
generalizations of these usual connections.


title: A symplectic proof of Verlinde factorization

abstract:
  We prove a multiplicity formula for Riemann-Roch numbers of reductions of
Hamiltonian actions of loop groups. This includes as a special case the
factorization formula for the quantum dimension of the moduli space of flat
connections over a Riemann surface.


title: Seiberg-Witten equations on tubes

abstract:
  We establish the adiabatic dissapearance of Seiberg-Witten tunnelings on
tubes R x N, where N is an S^1 fibration over a Riemann surface.


title: Harmonic morphisms with fibers of dimension one

abstract:
  I prove three classification results about harmonic morphisms whose fibers
have dimension one. All are valid when the domain is at least of dimension 4.
(The character of this overdetermined problem is very different when the
dimension of the domain is 3 or less.) The first result is a local
classification for such harmonic morphisms with specified target metric, the
second is a finiteness theorem for such harmonic morphisms with specified
domain metric, and the third is a complete classification of such harmonic
morphisms when the domain is a space form of constant sectional curvature
  The methods used are exterior differential systems and the moving frame. The
fundamental results are local, but, because of the rigidity of the solutions,
they allow a complete global classification.


title: Singular connection and Riemann theta function

abstract:
  We prove the Chern-Weil formula for SU(n+1)-singular connections over the
complement of an embedded oriented surface in smooth four manifolds. The
expression of the representation of a number as a sum of nonvanishing squares
is given in terms of the representations of a number as a sum of squares. Using
the number theory result, we study the irreducible SU(n+1)-representations of
the fundamental group of the complement of an embedded oriented surface in
smooth four manifolds.


title: The eta-form and a generalized Maslov index

abstract:
  We consider families of Dirac operators on the unit interval which depend on
parameters via boundary conditions. We study the associated eta forms and
Maslov cocyles. With this simple example we show how previous results of
Lesch/Woiciechowski and the first author on the eta invariant of cylinders
generalize to the family case.


title:
On the total curvature of minimal annuli in $R^3$ and Nitsche's
  conjecture

abstract:
  We present a proof of the generalized Nitsche's conjecture proposed by
W.H.Meeks III and H. Rosenberg: For $t\ge 0$, let $P_t$ denote the horizontal
plane of height $t$ over the $x_1,x_2$ plane. Suppose that $M \subset R^3$ is a
minimal annulus with the boundary contains in $P_0$ and that $M$ intersects
every $P_t$ in a simple closed curve. Then $M$ has finite total curvature. As a
consequence, we show that every properly embedded minimal surface of finite
topology in $R^3$ with more than one end has finite total curvature.


title: Moment maps and non-compact cobordisms

abstract:
  We define a moment map associated to a smooth torus action on a smooth
manifold, without a two-form. We define cobordisms of such structures, allowing
non compact manifolds as long as the moment maps are proper. We prove that a
compact manifold with a torus action and a moment map is cobordant to the
disjoint union of the normal bundles of the connected components of the fixed
points set. We use this to give simple new proofs of two formulas: Guillemin's
topological version of the abelian Jeffrey-Kirwan localization, and the
Guillemin-Lerman-Sternberg formula for the Duistermaat-Heckman measure.


title: Representatives of the Thom class of a vector bundle

abstract:
  After a review of several methods designed to produce equivariant cohomology
classes, we apply one introduced by Berline, Getzler and Vergne, to get a
family of representatives of the universal Thom class of a vector bundle.
Surprisingly, this family does not contain the representative given by
Matha\"{\i} and Quillen. However it contains a particularly simple and
symmetric representative that we construct explicitly.


title: Geometric zeta-functions on p-adic groups

abstract:
  Geometric zeta functions of Ihara and Hashimoto are generalized to higher
rank. The $p$-adic version of the Patterson conjecture is proven.


title:
Equivariant Holomorphic Morse Inequalities III: Non-Isolated Fixed
  Points

abstract:
  We prove the equivariant holomorphic Morse inequalities for a holomorphic
torus action on a holomorphic vector bundle over a compact Kahler manifold when
the fixed-point set is not necessarily discrete. Such inequalities bound the
twisted Dolbeault cohomologies of the Kahler manifold in terms of those of the
fixed-point set. We apply the inequalities to obtain relations of Hodge numbers
of the connected components of the fixed-point set and the whole manifold. We
also investigate the consequences in geometric quantization, especially in the
context of symplectic cutting.


title: Bott-type and equivariant Seiberg-Witten Floer homology I

abstract:
  We construct Bott-type and stable equivariant Seiberg-Witten Floer homology
and cohomology for rational homology spheres, and prove their diffeomorphism
invariance.


title:
Moduli Spaces of Stable Polygons and Symplectic Structures on
  $\bar{M}_{0,n}$

abstract:
  In this paper, certain natural and elementary polygonal objects in Euclidean
space, {\it the stable polygons}, are introduced, and the novel moduli spaces
${\bfmit M}_{{\bf r}, \epsilon}$ of stable polygons are constructed as complex
analytic spaces. Quite unexpectedly, these new moduli spaces are shown to be
projective and isomorphic to the moduli space $\bar{\cM}_{0,n}$ of the
Deligne-Mumford stable curves of genus 0. Further, built into the structures of
stable polygons are some natural data leading toward to a family of (classes
of) symplectic (K\"ahler) forms. To some degree, ${\bfmit M}_{{\bf r},
\epsilon}$ may be considered as symplectic counterparts of $\bar{\cM}_{0,n}$
and Kapranov's Chow quotient construction of $\bar{\cM}_{0,n}$. All these
together brings up a new tool to study the K\"ahler topology of
$\bar{\cM}_{0,n}$.


title: On Distortion and Thickness of Knots

abstract:
  What length of rope (of given diameter) is required to tie a particular knot?
To answer this question, we define some new notions of thickness for a space
curve, one based on Gromov's distortion, and another generalizing the thickness
of Litherland, Simon et al. We prove a basic inequality between these thickness
measures, and show that the distortion thickness is upper semi-continuous in
the C^0 topology, suggesting that shortest curves of thickness 1 should exist
in each knot class.


title:
Gluing formulae for Donaldson invariants for connected sums along
  surfaces

abstract:
  We solve a conjecture of Morgan and Szabo (Embedded genus 2 surfaces in
four-manifolds, Preprint) about the relationship of the basic classes of two
four-manifolds $X_i$ of simple type with $b_1=0$, $b^+>1$, such that there are
embedded Riemann surfaces of genus $g \geq 2$ and self-intersection zero (and
representing odd homology classes) with the basic classes of the manifold X
which appears as a connected sum along the surfaces (supposing this latter one
is of simple type). This is also expressed as constraints in the basic classes
of X. The result is in accordance with the results on Seiberg-Witten invariants
(Morgan, Szabo and Taubes, A product formula for the Seiberg-Witten invariants
and the generalized Thom conjecture).


title: A fake smooth CP^2 # RP^4

abstract:
  We show that the manifold *CP^2 # *RP^4, which is homotopy equivalent but not
homeomorphic to CP^2 # RP^4, is in fact smoothable.


title: Donaldson invariants for connected sums along surfaces of genus 2

abstract:
  We relate the Donaldson invariants of two four-manifolds $X_i$ with embedded
Riemann surfaces of genus 2 and self-intersection zero with the invariants of
the manifold X which appears as a connected sum along the surfaces. When the
original manifolds are of simple type with $b_1=0$ and $b^+>1$, X is of simple
type with $b_1=0$ and $b^+>1$ as well, and the relationship between the
invariants is expressed as constraints in the basic classes for X. Also we give
some applications. For instance, if $X_i$ have both $b_1=0$ then X is of simple
type with $b_1=0$, $b^+>1$, and has no basic classes evaluating zero on the
Riemann surface. Finally, we prove that any four-manifold with $b^+>1$ and with
an embedded surface of genus 2, self-intersection zero and representing an odd
homology class, is of finite type of second order.


title: H-projectively-equivalent four-dimensional Kahler manifolds

abstract:
  The paper is devoted to the investigation of four-dimensional Kahler
manifolds admitting non-affine H-projective mappings. We find all such
manifolds which are non-Einstein. In the paper also Kahler manifolds admitting
infinitesimal H-projective transformations are determined. It is proved that
the class of Kahler manifolds admitting H-projective mappings includes
generalized equidistant Kahler manifolds. The explicit expression for the
metrics of Ricci-flat four-dimensional generalized equidistant manifolds is
given.


title: Moduli spaces of PU(2)-monopoles

abstract:
  The goal of this article was the S^1-equivariant transversality-problem and
the compactification-problem for the moduli spaces of (perturbed)
PU(2)-monopoles. A substantially improved version entitled "Moduli spaces of
PU(2)-monopoles (revised version)" which gives simpler, clearer proofs of the
transversality results, has been published on arxiv in June 99 and appeared in
Asian J. Math, see Moduli spaces of PU(2)-Monopoles, Asian J. Math. Vol. 4, No.
2 (2000), 391-436.


title: Tight immersions and local differential geometry

abstract:
  An immersion of a compact manifold is tight if it admits the minimal total
absolute curvature over all immersions of the manifold. A prominent result in
the study of minimal total absolute curvature immersions is the theorem of
Chern and Lashof, which characterizes minimal total absolute curvature
immersions, and tight immersions, of spheres into a Euclidean space. In this
paper we examine tight immersions of highly connected manifolds; i.e.,
2k-dimensional manifolds that are (k-1)-connected by not k-connected, and
characterize the immersions of highest codimension.


title: Duality for Lie-Rinehart algebras and the modular class

abstract:
  We introduce a notion of duality for a Lie-Rinehart algebra giving certain
bilinear pairings in its cohomology generalizing the usual notions of
Poincar\'e duality in Lie algebra cohomology and de Rham cohomology. We show
that the duality isomorphisms can be given by a cap product with a suitable
fundamental class and hence may be taken natural in any reasonable sense.
Thereafter, for a Lie-Rinehart algebra satisfying duality, we introduce a
certain intrinsic module which provides a crucial ingredient for the
construction of the bilinear pairings. This module determines a certain class,
called modular class of the Lie-Rinehart algebra, which lies in a certain
Picard group generalizing the abelian group of flat line bundles on a smooth
manifold. Finally, we show that a Poisson algebra having suitable properties
determines a certain module for the corrresponding Lie-Rinehart algebra and
hence modular class whose square yields the module and characteristic class for
its Lie-Rinehart algebra mentioned before. In particular, this gives rise to
certain bilinear pairings in Poisson homology.


title:
Locally conformal flat Riemannian manifolds with constant principal
  Ricci curvatures and locally conformal flat C-spaces

abstract:
  It is proved that every locally conformal flat Riemannian manifold all of
whose Jacobi operators have constant eigenvalues along every geodesic is with
constant principal Ricci curvatures. A local classification (up to an isometry)
of locally conformal flat Riemannian manifold with constant Ricci eigenvalues
is given in dimensions 4,5,6,7 and 8. It is shown that any n-dimensional
$(4\leq n \leq 8)$ locally conformal flat Riemannian manifold with constant
principal Ricci curvatures is a Riemannian locally symmetric space.


title:
Continuous Families of Riemannian manifolds, isospectral on functions
  but not on 1-forms

abstract:
  The purpose of this paper is to present the first continuous families of
Riemannian manifolds isospectral on functions but not on 1-forms, and
simultaneously, the first continuous families of Riemannian manifolds with the
same marked length spectrum but not the same 1-form spectrum. The examples
presented here are Riemannian three-step nilmanifolds and thus provide a
counterexample to the Ouyang-Pesce Conjecture for higher-step nilmanifolds.
Ouyang and Pesce independently showed that all isospectral deformations of
two-step nilmanifolds must arise from the Gordon-Wilson method for constructing
isospectral nilmanifolds. In particular, all continuous families of Riemannian
two-step nilmanifolds that are isospectral on functions must also be
isospectral on p-forms for all p. They conjectured that all isospectral
deformations of nilmanifolds must arise in this manner. These examples arise
from a general method for constructing isospectral Riemannian nilmanifolds
previously introduced by the author.


title:
Continuous families of isospectral Riemannian metrics which are not
  locally isometric

abstract:
  Two Riemannian manifolds are said to be isospectral if the associated
Laplace-Belttrami operators have the same eigenvalue spectrum. If the manifolds
have boundary, one specifies DIrichlet or Neumann isospectrality depending on
the boundary conditions imposed on the eigenfunctions. We construct continuous
families of (Neumann and Dirichlet) isospectral metrics which have different
local geometry on manifolds with boundary in every dimension greater than 6 and
also new examples of pairs of closed isospectral manifolds with different local
geometry. These examples illustrate for the first time that the Ricci curvature
of a Riemannian manifold is not spectrally determined.


title: Kodaira Dimension and the Yamabe Problem

abstract:
  The Yamabe invariant Y(M) of a smooth compact manifold is roughly the
supremum of the scalar curvatures of unit-volume constant-scalar curvature
Riemannian metrics g on M. (To be absolutely precise, one only considers
constant-scalar-curvature metrics which are Yamabe minimizers, but this does
not affect the sign of the answer.) If M is the underlying smooth 4-manifold of
a complex algebraic surface (M,J), it is shown that the sign of Y(M) is
completely determined by the Kodaira dimension Kod (M,J). More precisely, Y(M)
< 0 iff Kod (M,J)=2; Y(M) = 0 iff Kod (M,J)=0 or 1; and Y(M) > 0 iff Kod (M,J)=
-infinity.


title: Differential Geometry of generalized almost quaternionic structures, 1

abstract:
  The fibre bundles adjoint to generalized almost quaternionic structures are
studied. The most important classes of generalized almost quaternionic
manifolds are considered.


title: Differential Geometry of generalized almost quaternionic structures, 2

abstract:
  Notions of self-dual and anti self-dual almost quaternionic structures are
introduced. The complete classification of self-dual and anti self-dual
generalized Kaehler manifolds is obtained.


title: Conformal invariant functionals of immersions of tori into R^3

abstract:
  We show, that higher analogs of the Willmore functional, defined on the space
of immersions M^2\rightarrow R^3, where M^2 is a two-dimensional torus, R^3 is
the 3-dimensional Euclidean space are invariant under conformal transformations
of R^3. This hypothesis was formulated recently by I.A.Taimanov
(dg-ga/9610013).
  Higher analogs of the Willmore functional are defined in terms of the
Modified Novikov-Veselov hierarchy. This soliton hierarchy is associated with
the zero-energy scattering problem for the two-dimensional Dirac operator.

categories: dg-ga hep-th math-ph math.DG math.MP nlin.SI solv-int


title:
The Variational Sequence on Finite Jet Bundle Extensions and the
  Lagrangian Formalism

abstract:
  The geometric Lagrangian theory (of arbitrary order) is based on the analysis
of some basic mathematical objects such as: the contact ideal, the (exact)
variational sequence, the existence of Euler-Lagrange and Helmholtz-Sonin
forms, etc. In this paper we give new and much simpler proofs for the whole
theory using Fock space methods. Using these results we give the most general
expression for a variationally trivial Lagrangian and the generic expression
for a locally variational system of partial differential equation.


title: Geometric Residue Theorems for Bundle Maps

abstract:
  In this paper we prove geometric residue theorems for bundle maps over a
compact manifold. The theory developed associates residues to the singularity
submanifolds of the map for any invariant polynomial. The theory is then
applied to a variety of settings: smooth maps between equidimensional
manifolds, CR-singularities, finite singularities and singularities of odd
forms as spinor bundle maps.


title: Dual Teichm\" uller spaces

abstract:
  We describe in elementary geometrical terms Teichm\" uller spaces of
decorated and holed surfaces. We construct explicit global coordinates on them
as well as on the spaces of measured laminations with compact and closed
support respectively. We show explicitly that the latter spaces are
asymptotically isomorphic to the former. We discuss briefly quantisation of
Teichm\" uller spaces and some other application of the constructed approach.
The paper does not require any preliminary knowledge of the subject above the
Poincar\' e uniformisation theorem.


title:
Conformally flat Einstein-like 4-manifolds and conformally flat
  Riemannian 4-manifolds all of whose Jacobi operators have parallel
  eigenspaces along every geodesic

abstract:
  A local classification of locally conformal flat Riemannian Einstein-like
four-manifolds as well as a local classification of all locally conformal flat
Riemannian four-manifolds for which all Jacobi operators have parallel
eigenspaces along every geodesic is given. Non-trivial explicit examples are
presented. The problem of local description of self-dual Einstein-like
four-manifolds is also treated. A complete explicit solution of the St\"ackel
system in dimension four is obtained.


title: Differential Geometry of Time-Dependent Mechanics

abstract:
  The usual formulations of time-dependent mechanics start from a given
splitting $Y=R\times M$ of the coordinate bundle $Y\to R$. From physical
viewpoint, this splitting means that a reference frame has been chosen.
Obviously, such a splitting is broken under reference frame transformations and
time-dependent canonical transformations. Our goal is to formulate
time-dependent mechanics in gauge-invariant form, i.e., independently of any
reference frame. The main ingredient in this formulation is a connection on the
bundle $Y\to R$ which describes an arbitrary reference frame. We emphasize the
following peculiarities of this approach to time-dependent mechanics. A phase
space does not admit any canonical contact or presymplectic structure which
would be preserved under reference frame transformations, whereas the canonical
Poisson structure is degenerate. A Hamiltonian fails to be a function on a
phase space. In particular, it can not participate in a Poisson bracket so that
the evolution equation is not reduced to the Poisson bracket. This fact becomes
relevant to the quantization procedure. Hamiltonian and Lagrangian formulations
of time-dependent mechanics are not equivalent. A degenerate Lagrangian admits
a set of associated Hamiltonians, none of which describes the whole mechanical
system given by this Lagrangian.


title: Gerstenhaber algebras and BV-algebras in Poisson geometry

abstract:
  The purpose of this paper is to establish an explicit correspondence between
various geometric structures on a vector bundle with some well-known algebraic
structures such as Gerstenhaber algebras and BV-algebras. Some applications are
discussed. In particular, we found an explicit connection between the
Koszul-Brylinski operator of a Poisson manifold and its modular class. As a
consequence, we prove that Poisson homology is isomorphic to Poisson cohomology
for unimodular Poisson structures.


title:
Space-time Averages in Macroscopic Gravity and Volume-preserving
  Coordinates

abstract:
  The definition of the covariant space-time averaging scheme for the objects
(tensors, geometric objects, etc.) on differentiable metric manifolds with a
volume n-form, which has been proposed for the formulation of macroscopic
gravity, is analyzed. An overview of the space-time averaging procedure in
Minkowski spacetime is given and comparison between this averaging scheme and
that adopted in macroscopic gravity is carried out throughout the paper. Some
new results concerning the algebraic structure of the averaging operator are
precisely formulated and proved, the main one being that the averaging bilocal
operator is idempotent iff it is factorized into a bilocal product of a
matrix-valued function on the manifold, taken at a point, by its inverse at
another point. The previously proved existence theorems for the averaging and
coordination bilocal operators are revisited with more detailed proofs of
related results. A number of new results concerning the structure of the
volume-preserving averaging operators and the class of proper coordinate
systems are given. It is shown, in particular, that such operators are defined
on an arbitrary n-dimensional differentiable metric manifold with a volume
n-form up to the freedom of (n-1) arbitrary functions of n arguments and 1
arbitrary function of (n-1) arguments. All the results given in this paper are
also valid whenever appropriate for affine connection manifolds including
(pseudo)-Riemannian manifolds.


title: Twisted product of Lie groups

abstract:
  In this article we define the twisted product of groups as the generalization
of the semidirect product of groups. We will find the necessary and sufficient
condition in order that the twisted product of groups to be a group. In
particular, for two copies of the same group, the twisted product of group by
itself through the action of inner automorphisms is a group if and only if the
initial group is a metabelian group. Further we will construct Lie algebra for
Lie group of a twisted product of Lie groups. In the case of twisted product of
Lie group by itself by means of the action of inner automorphisms we find the
dependence of the scalar curvature for resulting Lie group on the scalar
curvature for initial Lie group.


title: Correspondences, von Neumann algebras and holomorphic L^2 torsion

abstract:
  Given a holomorphic Hilbertian bundle on a compact complex manifold, we
introduce the notion of holomorphic $L^2$ torsion, which lies in the
determinant line of the twisted $L^2$ Dolbeault cohomology and represents a
volume element there. Here we utilise the theory of determinant lines of
Hilbertian modules over finite von Neumann algebras as developed in an earlier
paper by the authors. This specialises to the Ray-Singer-Quillen holomorphic
torsion in the finite dimensional case. We compute a metric variation formula
for the holomorphic $L^2$ torsion, which shows that it is not in general
independent of the choice of Hermitian metrics on the complex manifold and on
the holomorphic Hilbertian bundle, which are needed to define it. We therefore
initiate the theory of correspondences of determinant lines, that enables us to
define a relative holomorphic $L^2$ torsion for a pair of flat Hilbertian
bundles, which we prove is independent of the choice of Hermitian metrics on
the complex manifold and on the flat Hilbertian bundles.


title: The Kobayashi pseudodistance on almost complex manifolds

abstract:
  We extend the definition of the Kobayashi pseudodistance to almost complex
manifolds and show that its familliar properties are for the most part
preserved. We also study the automorphism group of an almost complex manifold
and finish with some examples.


title: A smooth counterexample to the Hamiltonian Seifert conjecture in R^6

abstract:
  A smooth counterexample to the Hamiltonian Seifert conjecture for
six-dimensional symplectic manifolds is found. In particular, we construct a
smooth proper function on the symplectic 2n-dimensional vector space, 2n > 4,
such that one of its non-singular level sets carries no periodic orbits of the
Hamiltonian flow. The function can be taken to be C^0-close and isotopic to a
positive-definite quadratic form so that the level set in question is isotopic
to an ellipsoid. This is a refinement of previously known constructions giving
such functions for 2n > 6. The proof is based on a new version of a symplectic
embedding theorem applied to the horocycle flow.


title:
On representation varieties of Artin groups, projective arrangements and
  the fundamental groups of smooth complex algebraic varieties

abstract:
  We prove that for any affine variety S defined over Q there exist Shephard
and Artin groups G such that a Zariski open subset U of S is biregular
isomorphic to a Zariski open subset of the character variety Hom(G,
PO(3))//PO(3). The subset U contains all real points of S . As an application
we construct new examples of finitely-presented groups which are not
fundamental groups of smooth complex algebraic varieties.


title: Classical Mechanics of Spinning Patricle in a Curved Space

abstract:
  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.


title: The Seiberg-Witten theory of homology 3-spheres

abstract:
  In this thesis we study the Seiberg-Witten theory of an oriented homology
3-sphere. The goal is to extract topological invariants - the Seiberg-Witten
invariants - by counting the solutions to the Seiberg-Witten equations on the
manifold. The first question we consider is whether the Seiberg-Witten
invariants depend on the geometric or analytic data involved in their
definition. In the first main result of this thesis, we completely determine
the dependence of the Seiberg-Witten invariants on the data involved in their
definition. In particular, we show that even for the simplest manifold, the
3-sphere $S^3$, the Seiberg-Witten invariants take infinitely many different
values.
  The rest of this thesis is devoted to understanding the Seiberg-Witten
invariants in a specific geometric setting - the surgery setting. In that
context we prove a gluing formula, which identifies the Seiberg-Witten
invariants as certain ``homological intersection numbers''.


title: Geometric Quantization and No Go Theorems

abstract:
  A geometric quantization of a K\"{a}hler manifold, viewed as a symplectic
manifold, depends on the complex structure compatible with the symplectic form.
The quantizations form a vector bundle over the space of such complex
structures. Having a canonical quantization would amount to finding a natural
(projectively) flat connection on this vector bundle. We prove that for a broad
class of manifolds, including symplectic homogeneous spaces (e.g., the sphere),
such connection does not exist. This is a consequence of a ``no go'' theorem
claiming that the entire Lie algebra of smooth functions on a compact
symplectic manifold cannot be quantized, i.e., it has no essentially nontrivial
finite-dimensional representations.


title: New Geometric Framework for SU(2) Gauge Theory

abstract:
  An explicit model of fiber bundle with local fibers being disinct copies of
vector 3-space is introduced. They are endowed with frames which are used as
local isotopic ones. The field local of isotopic frames is considered as gauge
field itself while the form of gauge connections is derived from it. A
covariant equation for the field of local frames is found. It is shown that
Yang-Mills equation follows from it, but variety of solutions of the new
equation is highly reduced in such that no ambiguities (Yang-Wu and vacuum
ones) arise. It is shown that Lagrangian for the field gives non-zero trace for
the stress-energy tensor and zero value for spin of the field of plane wave.
Some new solutions for the fields of punctual source and spherical wave are
found.


title: Noncommutative spectral geometry of Riemannian foliations

abstract:
  We construct spectral triples in a sense of noncommutative differential
geometry, associated with a Riemannian foliation on a compact manifold, and
describe its dimension spectrum.


title: Gromov Invariants and Symplectic Maps

abstract:
  Given a symplectomorphism f of a symplectic manifold X, one can form the
`symplectic mapping cylinder' $X_f = (X \times R \times S^1)/Z$ where the Z
action is generated by $(x,s,t)\mapsto (f(x),s+1,t)$. In this paper we compute
the Gromov invariants of the manifolds $X_f$ and of fiber sums of the $X_f$
with other symplectic manifolds. This is done by expressing the Gromov
invariants in terms of the Lefschetz zeta function of f and, in special cases,
in terms of the Alexander polynomials of knots. The result is a large set of
interesting non-Kahler symplectic manifolds with computational ways of
distinguishing them. In particular, this gives a simple symplectic construction
of the `exotic' elliptic surfaces recently discovered by Fintushel and Stern
and of related `exotic' symplectic 6-manifolds.


title: Geometry of growth: Approximation theorems for L^2 invariants

abstract:
  In this paper we study the problem of approximation of the $L^2$-topological
invariants by their finite dimensional analogues. We obtain generalizations of
the theorem of L\"uck, dealing with towers of finitely sheeted normal
coverings. We prove approximation theorems, establishing relations between the
homological invariants, corresponding to infinite dimensional representations
and sequences of finite dimensional representations, assuming that their
normalized characters converge. Also, we find an approximation theorem for
residually finite $p$-groups ($p$ is a prime), where we use the homology with
coefficients in a finite field $\fp$.
  We view sequences of finite dimensional flat bundles of growing dimension as
examples of growth processes. We study a von Neumann category with a Dixmier
type trace, which allows to describe the asymptotic invariants of growth
processes. We introduce a new invariant of torsion objects, the torsion
dimension. We show that the torsion dimension appears in general as an
additional correcting term in the approximation theorems; it vanishes under
some arithmeticity assumptions. We also show that the torsion dimension allows
to establish non-triviality of the Grothendieck group of torsion objects.


title:
The transversal wave equation and the noncommutative geodesic flow in
  Riemannian foliations

abstract:
  We study the transversal wave equation on a compact Riemannian foliated
manifold. As applications, we get an Egorov's type theorem for transversally
elliptic operators, state a relationship between the singularities of the
Fourier transform of the spectrum distribution function of a transversally
elliptic operator and the corresponding transversal bicharacteristic flow and
give a description of the noncommutative geodesic flow in transversal geometry
of Riemannian foliations.


title: Hamiltonian versus Lagrangian formulations of supermechanic

abstract:
  We take advantage of different generalizations of the tangent manifold to the
context of graded manifolds, together with the notion of super section along a
morphism of graded manifolds, to obtain intrinsic definitions of the main
objects in supermechanics such as, the vertical endomorphism, the canonical and
the Cartan's graded forms, the total time derivative operator and the
super--Legendre transformation. In this way, we obtain a correspondence between
the Lagrangian and the Hamiltonian formulations of supermechanics.


title: Sections along maps in Geometry and Physics

abstract:
  This paper points out the usefulness of the concept of derivation along a map
in many problems in Geometry and Physics. In particular it will be shown that
this approach allows us to translate the usual concepts arising in Geometrical
Mechanics, when appropriately written in a new way, to the framework of
Supermechanics.


title: Higher order Lagrangian supermechanics

abstract:
  Using supervector fields and graded forms along a morphism, we study the
geometry of ordinary differential superequations, extend the formalism of
higher order Lagrangian mechanics to the graded context and prove a
generalization of Noether's theorem.


title: Can We Look at The Quantisation Rules as Constraints?

abstract:
  In this paper we explore the idea of looking at the Dirac quantisation
conditions as $\hbar$-dependent constraints on the tangent bundle to
phase-space. Starting from the path-integral version of classical mechanics and
using the natural Poisson brackets structure present in the cotangent bundle to
the tangent bundle of phase- space, we handle the above constraints using the
standard theory of Dirac for constrained systems. The hope is to obtain, as
total Hamiltonian, the Moyal operator of time-evolution and as Dirac brackets
the Moyal ones. Unfortunately the program fails indicating that something is
missing. We put forward at the end some ideas for future work which may
overcome this failure.


title:
Closed curves in R^3: a characterization in terms of curvature and
  torsion, the Hasimoto map and periodic solutions of the Filament Equation

abstract:
  If a curve in R^3 is closed, then the curvature and the torsion are periodic
functions satisfying some additional constraints. We show that these
constraints can be naturally formulated in terms of the spectral problem for a
2x2 matrix differential operator. This operator arose in the theory of the
self-focusing Nonlinear Schrodinger Equation.
  A simple spectral characterization of Bloch varieties generating periodic
solutions of the Filament Equation is obtained. We show that the method of
isoperiodic deformations suggested earlier by the authors for constructing
periodic solutions of soliton equations can be naturally applied to the
Filament Equation.


title:
Eigenvalue Estimates for the Dirac Operator on Quaternionic Kaehler
  Manifolds

abstract:
  We consider the Dirac operator on compact quaternionic Kaehler manifolds and
prove a lower bound for the spectrum. This estimate is sharp since it is the
first eigenvalue of the Dirac operator on the quaternionic projective space.


title: Separated nets in Euclidean space and Jacobians of biLipschitz maps

abstract:
  We show that there are separated nets in the Euclidean plane which are not
biLipschitz equivalent to the integer lattice. The argument is based on the
construction of a continuous function which is not the Jacobian of a
biLipschitz map.


title:
Characteristic Classes for the Degenerations of Two-Plane Fields in Four
  Dimensions

abstract:
  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.


title: Killing spinors are Killing vector fields in Riemannian Supergeometry

abstract:
  A supermanifold M is canonically associated to any pseudo Riemannian spin
manifold (M_0,g_0). Extending the metric g_0 to a field g of bilinear forms
g(p) on T_p M, p\in M_0, the pseudo Riemannian supergeometry of (M,g) is
formulated as G-structure on M, where G is a supergroup with even part G_0\cong
Spin(k,l); (k,l) the signature of (M_0,g_0). Killing vector fields on (M,g)
are, by definition, infinitesimal automorphisms of this G-structure. For every
spinor field s there exists a corresponding odd vector field X_s on M. Our main
result is that X_s is a Killing vector field on (M,g) if and only if s is a
twistor spinor. In particular, any Killing spinor s defines a Killing vector
field X_s.


title: On Guichard's nets and Cyclic systems

abstract:
  In the first part, we give a self contained introduction to the theory of
cyclic systems in n-dimensional space which can be considered as immersions
into certain Grassmannians. We show how the (metric) geometries on spaces of
constant curvature arise as subgeometries of Moebius geometry which provides a
slightly new viewpoint. In the second part we characterize Guichard nets which
are given by cyclic systems as being Moebius equivalent to 1-parameter families
of linear Weingarten surfaces. This provides a new method to study families of
parallel Weingarten surfaces in space forms. In particular, analogs of Bonnet's
theorem on parallel constant mean curvature surfaces can be easily obtained in
this setting.


title:
Transports along Paths in Fibre Bundles. III. Consistency with Bundle
  Morphisms

abstract:
  The general problem for consistency between arbitrary transports along paths
in fibre bundles and bundle morphisms between them is formulated and
investigated. The special case of one fibre bundle, its morphism and transport
along paths acting in it is considered. The consistency between linear
transports along paths in a vector bundle and a Hermitian structure in it is
studied.


title: Lie-Rinehart algebras, Gerstenhaber algebras, and B-V algebras

abstract:
  For a Lie-Rinehart algebra (A,L), generators for the Gerstenhaber algebra
\Lambda_A L correspond bijectively to right (A,L)-connections on A in such a
way that B-V structures correspond to right (A,L)-module structures on A. When
L is projective as an A-module, given an exact generator \partial, the homology
of the B-V algebra (\Lambda_A L,\partial) coincides with that of L with
coefficients in A with respect to the right (A,L)-module structure determined
by \partial. When L is also of finite rank n, there are bijective
correspondences between (A,L)-connections on \Lambda_A^nL and right
(A,L)-connections on A and between left (A,L)- module structures on
\Lambda_A^nL and right (A,L)-module structures on A. Hence there are bijective
correspondences between (A,L)-connections on \Lambda_A^n L and generators for
the Gerstenhaber bracket on \Lambda_A L and between (A,L)-module structures on
\Lambda_A^n L and B-V algebra structures on \Lambda_A L. The homology of such a
B-V algebra (\Lambda_A L,\partial) coincides with the cohomology of L with
coefficients in \Lambda_A^n L, for the left (A,L)-module structure determined
by \partial. Some applications are discussed.


title: Quantum Hall Effect on the Hyperbolic Plane

abstract:
  In this paper, we study both the continuous model and the discrete model of
the Quantum Hall Effect (QHE) on the hyperbolic plane. The Hall conductivity is
identified as a geometric invariant associated to an imprimitivity algebra of
observables. We define a twisted analogue of the Kasparov map, which enables us
to use the pairing between $K$-theory and cyclic cohomology theory, to identify
this geometric invariant with a topological index, thereby proving the
integrality of the Hall conductivity in this case.


title: A Compactness Theorem for Invariant Connections

abstract:
  Necessary and sufficient conditions are given for the Palais-Smale Condition
C to hold for the Yang-Mills functional for invariant connections on a
principal bundle over a compact manifold of any dimension. It is assumed that
the connections are invariant under the action of a compact Lie group on the
manifold, and that all orbits of the action have codimension three or less.


title: A-hat Genus and Collapsing

abstract:
  We give relationships between the vanishing of the A-hat genus and the
possibility that a spin manifold can collapse with curvature bounded below.


title: Rozansky-Witten invariants via formal geometry

abstract:
  We show that recently constructed invariants of 3-dimensional manifolds and
of hyperkaehler manifolds (L.Rozansky and E.Witten, hep-th/9612216) come from
characteristic classes of foliations and from Gelfand-Fuks cohomology. In
particular, any symplectic foliation gives invariants of 3-manifolds.
  Our preprint has many intersections with the preprint alg-geom/9704009 by
M.Kapranov.

categories: dg-ga alg-geom hep-th math.AG math.DG math.QA q-alg


title: Seiberg-Witten Theory and Z/2^p actions on spin 4-manifolds

abstract:
  Furuta's ``10/8-th's'' theorem gives a bound on the magnitude of the
signature of a smooth spin 4-manifold in terms of the second Betti number. We
show that in the presence of a Z/2^p action, his bound can be strengthened. As
applications, we give new genus bounds on classes with divisibility and we give
a classification of involutions on rational cohomology K3's.
  We utilize the action of a twisted product of Pin(2) and Z/2^p on the
Seiberg-Witten moduli space. Our techniques also provide a simplification of
the proof of Furuta's theorem.


title:
Differential Rigidity of Anosov Actions of Higher Rank Abelian Groups
  and Algebraic Lattice Actions

abstract:
  We show that most homogeneous Anosov actions of higher rank Abelian groups
are locally smoothly rigid (up to an automorphism). This result is the main
part in the proof of local smooth rigidity for two very different types of
algebraic actions of irreducible lattices in higher rank semisimple Lie groups:
(i) the Anosov actions by automorphisms of tori and nil-manifolds, and (ii) the
actions of cocompact lattices on Furstenberg boundaries, in particular,
projective spaces. The main new technical ingredient in the proofs is the use
of a proper "non-stationary" generalization of the classical theory of normal
forms for local contractions.


title: Deformations of Nonholonomic Two-plane Fields in Four Dimensions

abstract:
  An Engel structure is a maximally non-integrable field of two-planes tangent
to a four-manifold. Any two such structures are locally diffeomorphic. We
investigate the space of global deformations of canonical Engel structures
arising out of contact three-manifolds. The main tool is Cartan's method of
prolongation and deprolongation which lets us pass back and forth between
certain Engel four-manifolds and contact three-manifolds. Every Engel manifold
inherits a natural one-dimensional foliation. Its leaves are the fibers of the
map from Engel to contact manifold, when this map exists. The foliation has a
transverse contact structure and tangential real projective structure. As an
application of our investigations, we show that a canonical Engel structure on
real projective three-space times an interval corresponds to geodesic flow on
the two-sphere, and that a subspace of its Engel deformations corresponds to
the space of Zoll metrics on the two-sphere.


title: Chaotic Geodesics in Carnot Groups

abstract:
  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''.


title: Radiation structures on isotropic hypersurfaces

abstract:
  The geometrical structures (in the sense of E. Cartan) are analyzed which
underlie the gravitational radiation phenomenon. Among the results are : - the
introduction of the adapted frame bundle to a congruence of isotropic
hypersurfaces in a Lorentzian manifold, - the description of the reduced frame
bundle which admits a unique radiation connection induced from the ambiant
space-time one, - the determination of the automorphisms of an integrable
radiation structure, - the repercussions of the geometry on the shape of the
stress-energy tensor in the Einstein's field equations.


title: Bounded and $L^2$ Harmonic Forms on Universal Covers

abstract:
  We relate the positivity of the curvature term in the Weitzenbock formula for
the Laplacian on p-forms on a complete manifold to the existence of bounded and
$L^2$ harmonic forms. In the case where the manifold is the universal cover of
a compact manifold, we obtain topological and geometric information about the
compact manifold. For example, we show that a compact manifold cannot admit one
metric with pinched negative curvature and another metric with positive
Weitzenbock term on two-forms. Many of these results can be thought of as
differential form analogues of Myers' theorem. We also give pinching conditions
on certain sums of sectional curvatures which imply the positivity of the
curvature term, and hence yield vanishing theorems. In particular, we construct
a compact manifold with planes of negative sectional curvature at each point
and which satisfies the hypothesis of our vanishing theorems.


title:
Variations of the boundary geometry of 3--dimensional hyperbolic convex
  cores

abstract:
  A fundamental object in a hyperbolic 3-manifold M is its convex core C(M),
defined as the smallest closed non-empty convex subset of M. We investigate the
way the geometry of the boundary S of C(M) varies as we vary the hyperbolic
metric of M. Thurston observed that the intrinsic metric of S is hyperbolic,
and that its bending is described by a measured geodesic lamination. We show
that, as the hyperbolic metric of the 3--manifold M varies differentiably, the
hyperbolic metric of the surface S varies in a C^1, but usually not C^2,
manner. Differentiability properties for the bending measured lamination are
conceptually less simple, because the space ML(S) of measured geodesic
laminations on S has no natural differentiable structure. However, ML(S) is a
piecewise linear manifold, and we also show that the bending measured geodesic
lamination varies differentiably in a piecewise linear sense.
  The two results are proved simultaneously, mixing the differentiable and
piecewise linear contexts. In particular, the 'corners' of the piecewise linear
structure of ML(S) account for the fact that the metric of S is not a C^2
function of the metric of M. A by-product of the proof is a similar
differentiablity property for Thurston's parametrization of complex projective
structures on a surface by hyperbolic metrics and measured laminations.


title: A Schlafli-type formula for convex cores of hyperbolic 3-manifolds

abstract:
  In 3-dimensional hyperbolic geometry, the classical Schlafli formula
expresses the variation of the volume of a hyperbolic polyhedron in terms of
the length of its edges and of the variation of its dihedral angles. We prove a
similar formula for the variation of the volume of the convex core of a
geometrically finite hyperbolic 3--manifold M, as we vary the hyperbolic metric
of M. In this case, the pleating locus of the boundary of the convex core is
not constant any more, but we showed in an earlier paper that the variation of
the bending of the boundary of the convex core is described by a geodesic
lamination with a certain transverse distribution. We prove that the variation
of the volume of the convex core is then equal to 1/2 the length of this
transverse distribution.


title: Uniformization of four manifolds

abstract:
  We characteristize those Einstein four manifolds which are locally symmetric
spaces of noncompact type. Namely they are four manifolds which admit solutions
to the (non-Abelian) Seiberg Witten equations and satisty certain
characterisitc number equality.


title: Holomorphic Morse Inequalities and Symplectic Reduction

abstract:
  We introduce Morse-type inequalities for a holomorphic circle action on a
holomorphic vector bundle over a compact Kaehler manifold. Our inequalities
produce bounds on the multiplicities of weights occurring in the twisted
Dolbeault cohomology in terms of the data of the fixed points and of the
symplectic reduction. This result generalizes both Wu-Zhang extension of
Witten's holomorphic Morse inequalities and Tian-Zhang Morse-type inequalities
for symplectic reduction.
  As an application we get a new proof of the Tian-Zhang relative index theorem
for symplectic quotients.


title: Symplectic aspects of the first eigenvalue

abstract:
  There are two themes in the present paper. The first one is spelled out in
the title, and is inspired by an attempt to find an analogue of Hersch-Yang-Yau
estimate for $lambda_1$ of surfaces in symplectic category. In particular we
prove that every split symplectic manifold $T^4 times M$ admits a compatible
Riemannian metric whose first eigenvalue is arbitrary large. On the other hand
for Kahler metrics compatible with a given integral symplectic form an upper
bound for $lambda_1$ does exist. The second theme is the study of Hamiltonian
symplectic fibrations over the 2-sphere. We construct a numerical invariant
called the size of a fibration which arises as the solution of certain
variational problems closely related to Hofer's geometry, K-area and coupling.
In some examples it can be computed with the use of Gromov-Witten invariants.
The link between these two themes is given by an observation that the first
eigenvalue of a Riemannian metric compatible with a symplectic fibration admits
a universal upper bound in terms of the size.


title: Invariants in c U p-Geometry

abstract:
  This note describes the construction of c U p-invariant differential
operators on statistical manifolds, i.e. of operators canonically associated to
a geometry which synthetizes the properties of conformal and projective
geometries.


title: Semidirect products and the Pukanszky condition

abstract:
  We study the general geometrical structure of the coadjoint orbits of a
semidirect product formed by a Lie group and a representation of this group on
a vector space. The use of symplectic induction methods gives new insight into
the structure of these orbits. In fact, each coadjoint orbit of such a group is
obtained by symplectic induction on some coadjoint orbit of a "smaller" Lie
group. We study also a special class of polarizations related to a semidirect
product and the validity of Pukanszky's condition for these polarizations. Some
examples of physical interest are discussed using the previous methods.


title:
An Extension of the Work of V. Guillemin on Complex Powers and Zeta
  Functions of Elliptic Pseudodifferential Operators

abstract:
  The purpose of this note is to extend the results of V. Guillemin on elliptic
self-adjoint pseudodifferential operators of order one, from operators defined
on smooth functions on a closed manifold to operators defined on smooth
sections in a vector bundle of Hilbert modules of finite type over a finite von
Neumann algebra.


title: On the Scalar Curvature of Einstein Manifolds

abstract:
  We show that there are high-dimensional smooth compact manifolds which admit
pairs of Einstein metrics for which the scalar curvatures have opposite signs.
These are counter-examples to a conjecture considered by Besse. The proof
hinges on showing that the Barlow surface has small deformations with ample
canonical line bundle.


title: Quantization of Presymplectic Manifolds and Circle Actions

abstract:
  We prove several versions of "quantization commutes with reduction" for
circle actions on manifolds that are not symplectic. Instead, these manifolds
possess a weaker structure, such as a spin^c structure. Our theorems work
whenever the quantization data and the reduction data are compatible; this
condition always holds if we start from a presymplectic (in particular,
symplectic) manifold.


title: Surfaces in 4-Manifolds

abstract:
  In this paper we introduce a technique, called rim surgery, which can change
a smooth embedding of an orientable surface of positive genus and nonnegative
self-intersection in a smooth 4-manifold while leaving the topological
embedding unchanged.


title: Local Properties of Self-Dual Harmonic 2-forms on a 4-Manifold

abstract:
  We will prove a Moser-type theorem for self-dual harmonic 2-forms on closed
4-manifolds, and use it to classify local forms on neighborhoods of singular
circles on which the 2-form vanishes. Removing neighborhoods of the circles, we
obtain a symplectic manifold with contact boundary - we show that the contact
form on each S^1\times S^2, after a slight modification, must be one of two
possibilities.


title: An Obstruction to Quantizing Compact Symplectic Manifolds

abstract:
  We prove that there are no nontrivial finite-dimensional Lie representations
of certain Poisson algebras of polynomials on a compact symplectic manifold.
This result is used to establish the existence of a universal obstruction to
quantizing a compact symplectic manifold, regardless of the dimensionality of
the representation.


title: Extensions of Lie-Rinehart algebras and the Chern-Weil construction

abstract:
  A Chern-Weil construction for extensions of Lie-Rinehart algebras is
introduced. This generalizes the classical Chern-Weil construction in
differential geometry and yields characteristic classes for arbitrary
extensions of Lie-Rinehart algebras. Some examples arising from spaces with
singularities and from foliations are given that cannot be treated by means of
the classical Chern-Weil construction.


title: The cohomology ring of polygon spaces

abstract:
  We compute the integer cohomology rings of the ``polygon spaces'' introduced
in [Hausmann,Klyachko,Kapovich-Millson]. This is done by embedding them in
certain toric varieties; the restriction map on cohomology is surjective and we
calculate its kernel using ideas from the theory of Gr\"obner bases. Since we
do not invert the prime 2, we can tensor with Z/2; halving all degrees we show
this produces the Z/2 cohomology rings of planar polygon spaces. In the
equilateral case, where there is an action of the symmetric group permuting the
edges, we show that the induced action on the integer cohomology is _not_ the
standard one, despite it being so on the rational cohomology [Kl]. Finally, our
formulae for the Poincar\'e polynomials are more computationally effective than
those known [Kl].


title:
Odd Invariant Semidensity and Divergence-like Operators on an Odd
  Symplectic Superspace

abstract:
  The divergence-like operator on an odd symplectic superspace which acts
invariantly on a specially chosen odd vector field is considered. This operator
is used to construct an odd invariant semidensity in a geometrically clear way.
The formula for this semidensity is similar to the formula of the mean
curvature of hypersurfaces in Euclidean space.


title: Surgery on Lagrangian and Legendrian Singularities

abstract:
  Let $\pi : E\to M$ be a smooth fiber bundle whose total space is a symplectic
manifold and whose fibers are Lagrangian. Let $L$ be an embedded Lagrangian
submanifold of $E$. In the paper we address the following question: how can one
simplify the singularities of the projection $\pi: L \to M$ by a Hamiltonian
isotopy of $L$ inside $E$?
  We give an answer in the case when $dim L = 2$ and both $L$ and $M$ are
orientable. A weaker version of the result is proved in the higher-dimensional
case. Similar results hold in the contact category.
  As a corollary one gets an answer to one of the questions of V.Arnold about
the four cusps on the caustic in the case of the Lagrangian collapse. As
another corollary we disprove Y.Chekanov's conjecture about singularities of
the Lagrangian projection of certain Lagrangian tori in $R^4$.


title: On the homotopy invariance of L^2 torsion for covering spaces

abstract:
  We prove the homotopy invariance of L^2 torsion for covering spaces, whenever
the covering transformation group is either residually finite or amenable. In
the case when the covering transformation group is residually finite and when
the L^2 cohomology of the covering space vanishes, the homotopy invariance was
established earlier by Lueck. We also give some applications of our results.


title: Duality of subanalytic sets

abstract:
  We study the link between a compact hypersurface in $\P^{n+1}$ and the set of
all its tangent planes. In this context, we identify $\P^{n+1}$ to the set of
linear subspaces of codimension one by orthogonal complementarity. This gives
rise to a kind of duality which has already been studied Bruce and
Romerro-Fuster, and relates a hypersurface to the set of its tangent planes.
But in these papers the dual, in this sense, of the set of tangent planes of a
hypersurface was not defined and iteration of the procedure was not possible.
Therefore we extend this type of duality to more general sets and achieve a
procedure which can be iterated and gives in fact an involution.


title: A frame bundle generalization of multisymplectic field theories

abstract:
  This paper presents a generalization of symplectic geometry to a principal
bundle over the configuration space of a classical field. This bundle, the
vertically adapted linear frame bundle, is obtained by breaking the symmetry of
the full linear frame bundle of the field configuration space, and it inherits
a generalized symplectic structure from the full frame bundle. The geometric
structure of the vertically adapted frame bundle admits vector-valued field
observables and produces vector-valued Hamiltonian vector fields, from which we
can define a Poisson bracket on the field observables. We show that the linear
and affine multivelocity spaces and multiphase spaces for geometric field
theories are associated to the vertically adapted frame bundle. In addition,
the new geometry not only generalizes both the linear and the affine models of
multisymplectic geometry but also resolves fundamental problems found in both
multisymplectic models.


title: Relative equilibria at singular points of the moment map

abstract:
  We prove a criterion for stability of relative equilibria in symmetric
Hamiltonian systems at singular points of the momentum map. This generalizes a
theorem of G.W. Patrick. The method of the proof is also useful in studying the
bifurcation of relative equilibria.


title: Quantization formula for singular reductions

abstract:
  In this note, we prove a quantization formula for singular reductions. The
main result is obtained as a simple application of an extended quantization
formula proved in [TZ2].