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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.
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.
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.
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.
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.
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.
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.
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.
We study the link between a compact hypersurface in $¶^{n+1}$ and the set of all its tangent planes. In this context, we identify $¶^{n+1}$ 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.
We construct the Seiberg-Witten theory on 3-manifolds with Euclidean ends (connected sums of $\R^3$ and a compact manifold) with perturbations which approximate $*dx_3$ at infinity, and describe the structure of the moduli spaces. The setup is inspired by Taubes's program of relating the 4-dimensional Seiberg-Witten invariant with `singular Gromov invariants' and has related applications.
We show that for a suitable class of ``Dirac-like'' operators there holds a Gluing Theorem for connected sums. More precisely, if $M_1$ and $M_2$ are closed Riemannian manifolds of dimension $n\ge 3$ together with such operators, then the connected sum $M_1 # M_2$ can be given a Riemannian metric such that the spectrum of its associated operator is close to the disjoint union of the spectra of the two original operators. As an application, we show that in dimension $n\equiv 3$ mod 4 harmonic spinors for the Dirac operator of a spin, $\Spinc$, or $\Spinh$ manifold are not topologically obstructed.
The strong maximum principle is proved to hold for weak (in the sense of support functions) sub- and super-solutions to a class of quasi-linear elliptic equations that includes the mean curvature equation for $C^0$ spacelike hypersurfaces in a Lorentzian manifold. As one application a Lorentzian warped product splitting theorem is given.
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.
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.
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.
Consider a lattice $Γ$ in a group $G = SL_2(\R), SO(1,n), SU(1,n)$, $SL_2(\Q_p)$. We discuss actions of $Γ$ by affine isometric transformations of Hilbert spaces. We show that for irreducible affine isometric action of $G$ its restriction to $Γ$ is irreducible. We prove the existence of canonical irreducible affine isometric actions of $Γ$ associated to actions of $Γ$ on $\R$- trees. Using such actions we construct irreducible representations of semigroup of probabilistic measures on $Γ$ and construct the series of representations of the groups of diffeomorphisms of Riemann surfaces enumerated by the points of Thurston compactification of Teichmüller (Teichmuller) space.
For $SU(2)$ (or $SO(3)$) Donaldson theory on a 4-manifold $X$, we construct a simple geometric representative for $μ$ of a point. Let $p$ be a generic point in $X$. Then the set $\{ [A] | F_A^-(p) $ is reducible $\}$, with coefficient -1/4 and appropriate orientation, is our desired geometric representative.
A general construction of an sh Lie algebra from a homological resolution of a Lie algebra is given. It is applied to the space of local functionals equipped with a Poisson bracket, induced by a bracket for local functions along the lines suggested by Gel'fand, Dickey and Dorfman. In this way, higher order maps are constructed which combine to form an sh Lie algebra on the graded differential algebra of horizontal forms. The same construction applies for graded brackets in field theory such as the Batalin-Fradkin-Vilkovisky bracket of the Hamiltonian BRST theory or the Batalin-Vilkovisky antibracket.