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This contains a list of (mostly very minor) corrections to the book Introduction to Symplectic Topology, Clarendon Press, Oxford, (1995), together with rewritten versions of two lemmas and some additional comments.
This note describes an invariant of rational homology 3-spheres in terms of configuration space integrals which in some sense lies between the invariants of Axelrod and Singer and those of Kontsevich.
We generalize Luck's Theorem to show that the L^2-Betti numbers of a residually amenable covering space are the limit of the L^2-Betti numbers of a sequence of amenable covering spaces. We show that any residually amenable covering space of a finite simplicial complex is of determinant class, and that the L^2 torsion is a homotopy invariant for such spaces. We give examples of residually amenable groups, including the Baumslag-Solitar groups.
The usual formulation of time-dependent mechanics implies a given splitting $Y=R\times M$ of an event space $Y$. This splitting, however, is broken by any time-dependent transformation, including transformations between inertial frames. The goal is the frame-covariant formulation of time-dependent mechanics on a bundle $Y\to R$ whose fibration $Y\to M$ is not fixed. Its phase space is the vertical cotangent bundle $V^*Y$ provided with the canonical 3-form and the corresponding canonical Poisson structure. An event space of relativistic mechanics is a manifold $Y$ whose fibration $Y\to R$ is not fixed.
We construct the first examples of continuous families of isospectral Riemannian metrics that are not locally isometric on closed manifolds, more precisely, on $S^n\times T^m$, where $T^m$ is a torus of dimension $m\ge 2$ and $S^n$ is a sphere of dimension $n\ge 4$. These metrics are not locally homogeneous; in particular, the scalar curvature of each metric is nonconstant. For some of the deformations, the maximum scalar curvature changes during the deformation.
Let $M$ be a compact Riemann surface, $h(x)$ a positive smooth function on $M$. In this paper, we consider the functional $$J(u)={1/2}\int \sb{M}|\bigtriangledown u|\sp 2 + 8π\int\sb{M}u -8π\log\int\sb{M}h\exp {u}$$. We give a sufficient condition under which $J$ achieves its minimum.
Extending work of Caffarelli-Yang and Tarantello, we present a variational existence proof for two-vortex solutions of the periodic Chern-Simons Higgs model and analyze the asymptotic behavior of these solutions as the parameter coupling the gauge field with the scalar field tends to 0.
Let L be a lattice in a connected Lie group. We show that besides a few exceptional cases, the deficiency of L is nonpositive.
We investigate the cross ratio for closed negatively curved manifolds. As one of several applications, we obtain that for two such homotopy equivalent manifolds M and N, the following is true : If M and N have the same marked length spectrum and if the Anosov splitting for M is C^1 then M and N have the same volume.
The analysis of mathematical structure of the method of operator manifold guides our discussion. The latter is a still wider generalization of the method of secondary quantization with appropriate expansion over the geometric objects. The nature of operator manifold provides its elements with both quantum field and geometry aspects, a detailed study of which is a subject of present paper. It yields a quantization of geometry differing in principle from all earlier suggested schemes.
We introduce the beta function of a knot in euclidean three-space. This is a meromorphic function of a complex variable which we prove admits a Bernstein type functional equation. We determine the first residues.
This research announcement gives a brief report of the main results in our paper "PU(2) monopoles, I: Regularity, Uhlenbeck compactness, and transversality" (Journal of Differential Geometry, to appear). We describe the existence of perturbations for the PU(2) monopole equations, yielding both useful transversality properties and an Uhlenbeck compactification for this perturbed moduli space.
Generalized Schouten, Froelicher-Nijenhuis and Froelicher-Richardson brackets are defined for an arbitrary Lie algebroid. Tangent and cotangent lifts of Lie algebroids are introduced and discussed and the behaviour of the related graded Lie brackets under these lifts is studied. In the case of the canonical Lie algebroid on the tangent bundle, a new common generalization of the Froelicher-Nijenhuis and the symmetric Schouten brackets, as well as embeddings of the Nijenhuis-Richardson and the Froelicher-Nijenhuis bracket into the Schouten bracket, are obtained.
The most important examples of a double vector bundle are provided by iterated tangent and cotangent functors: TTM, TT^*M, T^*TM, and T^*T^*M. We introduce the notions of the dual double vector bundle and the dual double vector bundle morphism. Theorems on canonical isomorphisms are formulated and proved. Several examples are given.
Using the Bryant representation, we define a new flux on homology classes of CMC-1 surfaces in hyperbolic 3-space, satisfying a balancing formula which is useful to show nonexistencd of certain kinds of complete CMC-1 surfaces.
Paper withdrawn - lemma 4.1 was false. Quite a lot of changes need to be made!
This paper studies the question of when a loop $ϕ$ in the group Symp$(M,ω)$ of symplectomorphisms of a symplectic manifold $(M,ω)$ is isotopic to a loop that is generated by a time-dependent Hamiltonian function. (Loops with this property are said to be Hamiltonian.) Our main result is that Hamiltonian loops are rigid in the following sense: if $ϕ$ is Hamiltonian with respect to $ω$, and if $ϕ'$ is a small perturbation of $ϕ$ that preserves another symplectic form $ω'$, then $ϕ'$ is Hamiltonian with respect to $ω'$. This allows us to get some new information on the structure of the flux group, i.e. the image of $π_1(Symp(M,ω))$ under the flux homomorphism. We give a complete proof of our result for some manifolds, and sketch the proof in general. The argument uses methods developed by Seidel for studying properties of Hamiltonian loops via the quantum homology of $M$.
The first group of differentiable cohomology of $\Diff(S^1)$, vanishing on the Möbius subgroup $PSL(2,R)\subset\Diff(S^1)$, with coefficients in modules of linear differential operators on $S^1$ is calculated. We introduce three non-trivial $PSL(2,R)$-invariant 1-cocycles on $\Diff(S^1)$ generalizing the Schwarzian derivative.