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arXiv1997

Comparison and Rigidity Theorems in Semi-Riemannian Geometry

The comparison theory for the Riccati equation satisfied by the shape operator of parallel hypersurfaces is generalized to semi-Riemannian manifolds of arbitrary index, using one-sided bounds on the Riemann tensor which in the Riemannian case correspond to one-sided bounds on the sectional curvatures. Starting from 2-dimensional rigidity results and using an inductive technique, a new class of gap-type rigidity theorems is proved for semi-Riemannian manifolds of arbitrary index, generalizing those first given by Gromov and Greene-Wu. As applications we prove rigidity results for semi-Riemannian manifolds with simply connected ends of constant curvature.

dg-ga↗
arXiv1997

Singular Reduction and Quantization

Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The ``quantization commutes with reduction'' theorem asserts that the G-invariant part of the equivariant index of M is equal to the Riemann-Roch number of the symplectic quotient of M, provided the quotient is nonsingular. We extend this result to singular symplectic quotients, using partial desingularizations of the symplectic quotient to define its Riemann-Roch number. By similar methods we also compute multiplicities for the equivariant index of the dual of a prequantum bundle, and furthermore show that the arithmetic genus of a Hamiltonian G-manifold is invariant under symplectic reduction.

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arXiv1997

Fedosov Manifolds

In this paper we study geometry of symmetric torsion-free connections which preserve a given symplectic form

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arXiv2000

The surfaces capable of division into Infinitesimal Squares by their Curves of Curvature

Classically, isothermic surfaces are characterized as those surfaces which are "divisible into infinitesimal squares by their curvature lines". This characterization is the direct analogue to the definition of discrete isothermic nets. In order to understand the relations between the discrete and the smooth theory better, it is described how to give the classical characterization a rigorous meaning in the sense of modern differential geometry.

dg-ga↗
arXiv1997

Exterior differential systems: a geometric approach to PDE. Lecture notes from the 1997 Daewoo workshop

This is an elementary introduction to exterior differential systems motivated by two examples: minimal submanifolds and the isometric embedding problem. The two main goals of the lectures are: 1. To explain how to find an appropriate geometric setting for studying a given system of pde. 2. To explain the Cartan algorithm to determine the moduli space of local solutions to any given exterior differential system and an appropriate initial value problem for the system.

dg-ga↗
arXiv2000

Pseudo-holomorphic curves and the Weinstein conjecture

We established existence of periodic Reeb orbits for a large class of tight contact structures on closed 3-manifolds, notably the Stein fillable structures, based on a fundamental theorem of Cliff Taubes on symplectic 4-manifolds.

dg-ga↗
arXiv1997

Tree-like curves and their inflection points

We give a criterion when a planar tree-like curve, i.e. a generic immersed plane curve each double point of which cuts it into two disjoint parts, can be send by a diffeomorphism of the plane onto a curve with no inflection points. We also present some upper and lower bounds for the minimal number of inflection points on such curves unremovable by diffeomorphisms of the plane in terms of their combinatorics.

dg-ga↗
arXiv1999

On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols

We study various aspects of the noncommutative residue for an algebra of pseudodifferential operators whose symbols have an expansion $a\sim \sum_{j=0}^\infty a_{m-j}, a_{m-j}(x,ξ)=\sum_{l=0}^k a_{m-j,l}(x,ξ) \log^l|ξ|,$ where $a_{m-j,l}$ is homogeneous in $ξ$ of degree $m-j$. We will explain why this algebra of pseudodifferential operators is natural. For a pseudodifferential operator in this class, $A$, and a classical elliptic pseudodifferential operator, $P$, we show that the generalized zeta-function $\Tr(AP^{-s})$ has a meromorphic continuation to the whole complex plane, however possibly with higher order poles. Our algebra of operators has a bigrading given by the order and the highest log-power occuring in the symbol expansion. We construct "higher" noncommutative residue functionals on the subspaces given by the log-grading. However, in contrast to the classical case we prove that the whole algebra does not admit any nontrivial traces. Finally we show that the analogue of the Kontsevich-Vishik trace also exists on our algebra. Our method also provides an alternative approach to the Kontsevich-Vishik trace.

dg-ga↗
arXiv1997

Invariants of Velocities, and Higher Order Grassmann Bundles

An $(r,n)$-velocity is an $r$-jet with source at $0 \in \R^n$, and target in a manifold $Y$. An $(r,n)$-velocity is said to be regular, if it has a representative which is an immersion at $0 \in \R^{n}$. The manifold $T^{r}_{n}Y$ of $(r,n)$-velocities as well as its open, $L^{r}_{n}$-invariant, dense submanifold $\Imm T^{r}_{n}Y$ of regular $(r,n)$-velocities, are endowed with a natural action of the differential group $L^{r}_{n}$ of invertible $r$-jets with source and target $0 \in \R^{n}$. In this paper, we describe all continuous, $L^{r}_{n}$-invariant, real-valued functions on $T^{r}_{n}Y$ and $\Imm T^{r}_{n}Y$. We find local bases of $L^{r}_{n}$-invariants on $\Imm T^{r}_{n}Y$ in an explicit, recurrent form. To this purpose, higher order Grassmann bundles are considered as the corresponding quotients $P^{r}_{n}Y = \Imm T^{r}_{n}Y/L^{r}_{n}$, and their basic properties are studied. We show that nontrivial $L^{r}_{n}$-invariants on $\Imm T^{r}_{n}Y$ cannot be continuously extended onto $T^{r}_{n}Y$.

dg-ga↗
arXiv1997

Ricci curvature, minimal surfaces and sphere theorems

Using an analogue of Myers' theorem for minimal surfaces and three dimensional topology, we prove the diameter sphere theorem for Ricci curvature in dimension three and a corresponding eigenvalue pinching theorem. This settles these two open problems for closed 3 manifolds with positive Ricci curvature since they are both false in dimensions greater than three.

dg-ga↗
arXiv1997

Kahler manifolds with large isometry group

We investigate compact Kahler manifolds, which are acted on by a semisimple compact Lie group G of isometries with one hypersurface orbit. In case of ordinary action and projectable complex structure, we set up a one to one correspondence between such manifolds and abstract models. The Ricci tensor is then computed and we fully characterize the Kahler-Einstein manifolds with an ordinary cohomogeneity one action and projectable complex structure.

dg-ga↗
arXiv1997

continious cohomology of groups of volume-preserving and symplectic diffepmorphisms, measurable transfer and higher asymtotic cycles

I construct the real counterparts (which I call Borel-Bott classes) of the R/Z classes constructed in "Characteristic classes in symplectic topology", to appear, in the cohomology of volume-preserving and symplectomorhisms of a compact (symplectic) manifold.I show that, for the symplectic action of the mapping class group in the moduli space of stable vector bundles over a Riemann surface, the restriction of the first constructed class from the symplectomorphism group gives a generator for the second (bounded) cohomology of the mapping class group.

dg-ga↗
arXiv1998

From a Relativistic Phenomenology of Anyons to a Model of Unification of Forces via the Spencer Theory of Lie Structures

Starting from a relativistic phenomenology of anyons in crystals, we discuss the concept of relativistic interaction and the need to unify electromagnetism and gravitation within the Spencer cohomology of Lie equations. Then, from the sophisticated non-linear Spencer complex of the Poincaré and conformal Lie pseudogroups, we build up a non-linear relative complex assigned to a gauge sequence for electromagnetic and gravitational potentials and fields. Then, using a conformally equivariant Lagrangian density, we deduce, first, the two first steps of its corresponding Janet complex and second, the dual relative linear complex. We conclude by giving suggestions for higher unification with the weak and strong interactions and interpretations of the Lagrangian density as a thermodynamical function and quantum wave-function.

dg-ga↗
arXiv1997

Deformations of representations of fundamental groups of open Kaehler manifolds

Given a compact Kaehler manifold, we consider the complement U of a divisor with normal crossings. We study the variety of unitary representations of the fundamental group of U with certain restrictions related to the divisor. We show that the possible singularities of this variety as well as of the corresponding moduli space of irreducible representations are quadratic. In the course of our proof we exhibit a differential graded Lie algebra of which reflects our deformation problem.

dg-ga↗
arXiv1998

Transports along maps in fibre bundles

Generalized are the investigated in other works of the author transports along paths in fibre bundles to transports along arbitrary maps in them. Their structure and some properties are studied. Special attention is paid to the linear case and the case when the map's domain is a Cartesian product of two sets. Also considered are the consistency with bundle morphisms and a number of special cases.

dg-ga↗
arXiv1997

Investigation and application of the dressing action on surfaces of constant mean curvature

We investigate the dressing action on surfaces of constant mean curvature (CMC surfaces) in Euclidean space. In particluar, we show that for CMC surfaces with umbilics the isotropy group under dressing is always trivial. This result is applied to the investigation of CMC surfaces whose metric is invariant under a group of automorphisms of the parameter domain.

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