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It is shown that moduli spaces of complete families of compact complex hypersurfaces in complex manifolds often come equipped canonically with projective structures satisfying some natural integrability conditions.
A systematic description of the Wess-Zumino-Witten model is presented. The symplectic method plays the major role in this paper and also gives the relationship between the WZW model and the Chern-Simons model. The quantum theory is obtained to give the projective representation of the Loop group. The Gauss constraints for the connection whose curvature is only focused on several fixed points are solved. The Kohno connection and the Knizhnik-Zamolodchikov equation are derived. The holonomy representation and $\check R$-matrix representation of braid group are discussed.
This is a note for the conference proceedings Topological and Geometrical Problems related to Quantum Field Theory. We summarize our joint work with Dai about eta invariants on manifolds with boundary. Then we apply these results to prove the curvature and holonomy formulas for the natural connection on the determinant line bundle of a family of Dirac operators. These were originally proved by Bismut and the author--the proofs here are much simpler.
Examples of Morse functions with integrable gradient flows on some classical Riemannian manifolds are considered. In particular, we show that a generic height function on the symmetric embeddings of classical Lie groups and certain symmetric spaces is a perfect Morse function, i.e. has as many critical points as the homology requires, and the corresponding gradient flow can be described explicitly. This gives an explicit cell decomposition and geometric realization of the homology for such a manifold. As another application of the integrable Morse functions we give an elementary proof of Vassiljev's theorem on the flag join of Grassmannians.
We show that Tolman's example (of a six dimensional Hamiltonian $T^2$-space with isolated fixed points and no compatible Kähler structure) can be constructed from the flag variety $U(3)/U(1)^3$ by $U(2)$-equivariant symplectic surgery. This implies that Tolman's space has a ``transversal multiplicity-free'' action of $U(2)$ and that Delzant's theorem ``every compact multiplicity-free torus action is Kähler'' \cite{D1} does not generalize to non-abelian actions.
We address two fundamental and well-known problems of Gromov and Lyndon: \demo{Problem A} (Gromov, see [5]). Consider a category $M_n$ of closed manifolds of dimension $n$ with nonzero-degree ways as morphisms. Study a partial order $M \ge N \Leftrightarrow Mor (M, N) \neq ϕ$. For which $N$ the degrees of maps $f: M \to N$ are bounded for all $M$? \demo{Problem B} (Lyndon, [12], problem 13). Extend and relate the theories of deficiency, the rate of growth and the Euler-Poincaré characteristic. In particular, what influence does the deficiency have on the structure of an infinite group?
A new approach is proposed for study structure and properties of the total squared mean curvature $W$ of surfaces in ${\bf R}^3$. It is based on the generalized Weierstrass formulae for inducing surfaces. The quantity $W$ (Willmore functional) is shown to be invariant under the modified Novikov--Veselov hierarchy of integrable flows. The $1+1$--dimensional case and, in particular, Willmore tori of revolution, are studied in details. The Willmore conjecture is proved for the mKDV--invariant Willmore tori.
We consider compact complex surfaces with Hermitian metrics which are Einstein but not Kaehler. It is shown that the manifold must be CP2 blown up at 1,2, or 3 points, and the isometry group of the metric must contain a 2-torus. Thus the Page metric on CP2#(-CP2) is almost the only metric of this type.
A membrane technique, in which the symplectic and Ricci forms are integrated over surfaces in a complexification of the phase space, as well a ``creation" connection with zero curvature over lagrangian submanifolds, is used to obtain a unified quantization including a noncommutative algebra of functions, its representations, the Dirac axioms, coherent integral transformations for solutions of spectral and Cauchy problems, and trace formulas. (To be published in Russian Journal of Mathematical Physics,1995,v.3,n.3)
We generalize the Novikov inequalities for 1-forms in two different directions: first, we allow non-isolated critical points (assuming that they are non-degenerate in the sense of R.Bott), and, secondly, we strengthen the inequalities by means of twisting by an arbitrary flat bundle. The proof uses Bismut's modification of the Witten deformation of the de Rham complex; it is based on an explicit estimate on the lower part of the spectrum of the corresponding Laplacian. In particular, we obtain a new analytic proof of the degenerate Morse inequalities of Bott.
In this paper we show that if the limit set is not small ,marked length spectrum determines geometric structure of rank one locally symmetric manifolds.
We generalize the Novikov inequalities for 1-forms in two different directions: first, we allow non-isolated critical points (assuming that they are non-degenerate in the sense of R.Bott), and, secondly, we strengthen the inequalities by means of twisting by an arbitrary flat bundle. We also obtain an $L^2$ version of these inequalities with finite von Neumann algebras. The proof of the main theorem uses Bismut's modification of the Witten deformation of the de Rham complex; it is based on an explicit estimate on the lower part of the spectrum of the corresponding Laplacian.
We prove that Riemannian metrics with a uniform weak norm can be smoothed to having arbitrarily high regularity. This generalizes all previous smoothing results. As a consequence we obtain a generalization of Gromov's almost flat manifold theorem. A uniform Betti number estimate is also obtained.
This paper is a review of the twistor theory of irreducible G-structures and affine connections. Long ago, Berger presented a very restricted list of possible irreducibly acting holonomies of torsion-free affine connections. His list was complete in the part of metric connections, while the situation with holonomies of non-metric torsion-free affine connections was and remains rather unclear. One of the results discussed in this review asserts that any torsion-free holomorphic affine connection with irreducibly acting holonomy group can, in principle, be constructed by twistor methods. Another result reveals a new natural subclass of affine connections with "very little torsion" which shares with the class of torsion-free affine connections two basic properties --- the list of irreducibly acting holonomy groups of affine connections in this subclass is very restricted and the links with the twistor theory are again very strong.
The Einstein/Abelian-Yang-Mills Equations reduce in the stationary and axially symmetric case to a harmonic map with prescribed singularities $\p\colon\R^3\smΣ\to\H^{k+1}_\C$ into the $(k+1)$-dimensional complex hyperbolic space. In this paper, we prove the existence and uniqueness of harmonic maps with prescribed singularities $\p\colon\R^n\smΣ\to\H$, where $Σ$ is an unbounded smooth closed submanifold of $\R^n$ of codimension at least $2$, and $\H$ is a real, complex, or quaternionic hyperbolic space. As a corollary, we prove the existence of solutions to the reduced stationary and axially symmetric Einstein/Abelian-Yang-Mills Equations.
Suppose that the initial triangle formed by the three moving masses of the three-body problem is similar to the triangle formed at some later time. We derive a simple integral formula for the overall rotation relating the two triangles. The formula is based on the fact that the space of similarity classes of triangles forms a two-sphere which we call the shape sphere. The formula consists of a ``dynamic'' and ``geometric'' term. The geometric term is the integral of a universal two-form on a``reduced configuration space''. This space is a two-sphere bundle over the shape sphere. The fibering spheres are instantaneous versions of the angular momentum sphere appearing in rigid body motion. Our derivation of the formula is similar in spirit to our earlier reconstruction formula for the rigid body motion.
Let $G$ be a Lie group, with an invariant non-degenerate symmetric bilinear form on its Lie algebra, let $π$ be the fundamental group of an orientable (real) surface $M$ with a finite number of punctures, and let $\bold C$ be a family of conjugacy classes in $G$, one for each puncture. A finite-dimensional construction used earlier to obtain a symplectic structure on the moduli space of flat $G$-bundles over compact $M$ is extended to the punctured case. It yields a symplectic structure on a certain smooth manifold $\Cal M_{\bold C}$ containing the space $\roman{Hom}(π,G)_{\bold C}$ of homomorphisms mapping the generators corresponding to the punctures into the corresponding conjugacy classes. It also yields a Hamiltonian $G$-action on $\Cal M_{\bold C}$ such that the reduced space equals the moduli space $\roman{Rep}(π,G)_{\bold C}$ of representations. For $G$ compact, each such space, obtained by finite-dimensional symplectic reduction, is a {\it stratified symplectic space\/}. For $G=U(n)$ one gets moduli spaces of semistable holomorphic parabolic bundles or spaces closely related to them.
We discuss Ghys' theorem on 4 zeroes of the Schwarzian derivative and its relation with flattening points of Legendrian curves and Sturm theory.