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We consider real isotropic geodesics on manifolds endowed with a pseudoconformal structure and their applications to the theory of lightlike hypersurfaces on such manifolds, the geometry of four-dimensional conformal structures of Lorentzian type, and a classification of the Einstein spaces. | On conformal invariance of isotropic geodesics | 14,200 |
The Novikov-Shubin invariants for a non-compact Riemannian manifold M can be defined in terms of the large time decay of the heat operator of the Laplacian on square integrable p-forms on M. For the (2n+1)-dimensional Heisenberg group H, the Laplacian can be decomposed into operators in the conjugate of the generalised Bargmann representations which, when restricted to the centre of H, are characters. The representation space is an anti-Fock space of anti-holomorphic functions on complex n-space which are square integrable with respect to a Gaussian measure. In this paper, the eigenvalues of decomposed operators are calculated, using operators which commute with the Laplacian; this information determines all the Novikov-Shubin invariants of H. Further, some eigenvalues of operators connected with nilpotent Lie groups of Heisenberg type are calculated in the later sections. | The Laplacian on $p$-forms on the Heisenberg group | 14,201 |
A representation of generalized Weierstrass formulae for an immersion of generic surfaces into a 4-dimensional complex space in terms of spinors treated as minimal left ideals of Clifford algebras is proposed. The relation between integrable deformations of surfaces via mVN-hierarchy and integrable deformations of spinor fields on the surface is also discussed. | Generalized Weierstrass representation for surfaces in terms of
Dirac-Hestenes spinor field | 14,202 |
We introduce a 1-cocycle on the group of diffeomorphisms Diff$(M)$ of a smooth manifold $M$ endowed with a projective connection. This cocycle represents a nontrivial cohomology class of $\Diff(M)$ related to the Diff$(M)$-modules of second order linear differential operators on $M$. In the one-dimensional case, this cocycle coincides with the Schwarzian derivative, while, in the multi-dimensional case, it represents its natural and new generalization. This work is a continuation of \cite{bo} where the same problems have been treated in one-dimensional case. | Schwarzian derivative related to modules of differential operators on a
locally projective manifold | 14,203 |
We study the Yamabe invariant of manifolds obtained as connected sums along submanifolds of codimension greater than 2. In particular, given a compact smooth manifold M which does not admit metrics of positive scalar curvature, we prove that the Yamabe invariant of M is an upper bound for the Yamabe invariant of any manifold obtained by performing surgery in M on spheres of codimension greater than 2 . | Surgery and the Yamabe invariant | 14,204 |
For a compact connected manifold M of dimension n greater than 3 and with no metric of positive scalar curvature, we prove that the Yamabe invariant is unchanged under surgery on spheres of dimension different from 1, n-2 and n-1. We use this result to give new computations of the invariant in dimension four and display new examples of 4-manifolds which do not admit Einstein metrics. | Computations of the Yamabe invariant | 14,205 |
We prove that the Yamabe invariant of any simply connected smooth manifold of dimension n greater than four is non-negative. Equivalently that the infimum of the L^{n/2} norm of the scalar curvature, over the space of all Riemannian metrics on the manifold, is zero. | The Yamabe invariant of simply connected manifolds | 14,206 |
We construct closed symplectic manifolds for which spherical classes generate arbitrarily large subspaces in 2-homology, such that the first Chern class and cohomology class of the symplectic form both vanish on all spherical classes. We construct both Kaehler and non-Kaehler examples, and show independence of the conditions that these two cohomology classes vanish on spherical homology. In particular, we show that the symplectic form can pair trivially with all spherical classes even when the Chern class pairs nontrivially. | On symplectically aspherical manifolds with nontrivial pi_2 | 14,207 |
We define a general notion of abstract double Lie algebroid. We show (1) that the double Lie algebroid of a double Lie groupoid is a double Lie algebroid in this sense; (2) that the double cotangent constructed from Lie algebroid structures on a vector bundle A and its dual A* is a double Lie algebroid if and only if (A, A*) is a Lie bialgebroid; (3) that a vacant double Lie algebroid structure is equivalent to a matched pair structure on the side Lie algebroids. | Double Lie algebroids and the double of a Lie bialgebroid | 14,208 |
Classification results are given for (i) compact quaternionic K\"ahler manifolds with a cohomogeneity-one action of a semi-simple group, (ii) certain complete hyperK\"ahler manifolds with a cohomogeneity-two action of a semi-simple group preserving each complex structure, (iii) compact 3-Sasakian manifolds which are cohomogeneity one with respect to a group of 3-Sasakian symmetries. Information is also obtained about non-compact quaternionic K\"ahler manifolds of cohomogeneity one and the cohomogeneity of adjoint orbits in complex semi-simple Lie algebras. | Quaternionic Kähler Manifolds of Cohomogeneity One | 14,209 |
Quantization needs evaluation of all of states of a quantized object rather than its stationary states with respect to its energy. In this paper, we have investigated moduli $\CMeP$ of a quantized elastica, a quantized loop with an energy functional associated with the Schwarz derivative, on a Riemann sphere $\PP$. Then it is proved that its moduli space is decomposed to a set of equivalent classes determined by flows obeying the Korteweg-de Vries (KdV) hierarchy which conserve the energy. Since the flow obeying the KdV hierarchy has a natural topology, it induces topology in the moduli space $\CMeP$. Using the topology, $\CMeP$ is classified. Studies on a loop space in the category of topological spaces $\Top$ are well-established and its cohomological properties are well-known. As the moduli space of a quantized elastica can be regarded as a loop space in the category of differential geometry $\DGeom$, we also proved an existence of a functor between a triangle category related to a loop space in {\bf Top} and that in $\DGeom$ using the induced topology. As Euler investigated the elliptic integrals and its moduli by observing a shape of classical elastica on $\CC$, this paper devotes relations between hyperelliptic curves and a quantized elastica on $\PP$ as an extension of Euler's perspective of elastica. | On the Moduli of a quantized loop in P and KdV flows: Study of
hyperelliptic curves as an extension of Euler's perspective of elastica I | 14,210 |
The main results of our paper deal with the lifting problem for multilinear differential operators between complexes of horizontal de Rham forms on the infinite jet bundle. We answer the question when does an n-multilinear differential operator from the space of (N,0)-forms (where N is the dimension of the base) to the space of (N-s,0)-forms allow an n-multilinear extension of degree (-s,0) defined on the whole horizontal de Rham complex. To study this problem we define a differential graded operad DEnd of multilinear differential endomorphisms, which we prove to be acyclic in positive degrees (negative mapping degrees) and describe the cohomology group in degree zero in terms of the characteristic. As a corollary, we solve the lifting problem. An important application to mathematical physics is the proof of existence of a strongly homotopy Lie algebra structure extending a Lie bracket on the space of functionals. | Differential Operator Endomorphisms of an Euler-Lagrange Complex | 14,211 |
Goresky, Kottwitz and MacPherson have recently shown that the computation of the equivariant cohomology ring of a G-manifold can be reduced to a computation in graph theory. This opens up the possibility that many of the fundamental theorems in equivariant de Rham theory may, on closer inspection, turn out simply to be theorems about graphs. In this paper we show that for some familiar theorems, this is indeed the case. | Equivariant de Rham Theory and Graphs | 14,212 |
In 1968, Milnor conjectured that a complete noncompact manifold with nonnegative Ricci curvature has a finitely generated fundamental group. The author applies the Excess Theorem of Abresch and Gromoll (1990), to prove two theorems. The first states that if such a manifold has small linear diameter growth then its fundamental group is finitely generated. The second states that if such a manifold has an infinitely generated fundamental group then it has a tangent cone at infinity which is not polar. A corollary of either theorem is the fact that if such a manifold has linear volume growth, then its fundamental group is finitely generated. | Nonnegative Ricci curvature, small linear diameter growth and finite
generation of fundamental groups | 14,213 |
We study a sequence of connections which is associated with a Riemannian metric and an almost symplectic structure on a manifold. We prove that if this sequence is trivial (i.e. constant) or 2-periodic, then the manifold has a canonical K\"ahler structure. | A sequence of connections and a characterization of Kähler manifolds | 14,214 |
We study the behavior of the Yang-Mills flow for unitary connections on compact and non-compact oriented surfaces with varying metrics. The flow can be used to define a one dimensional foliation on the space of SU(2) representations of a once punctured surface. This foliation universalizes over Teichm\"uller space and is equivariant with respect to the action of the mapping class group. It is shown how to extend the foliation as a singular foliation over the Strebel boundary of Teichm\"uller space, and continuity of this extension is the main result of the paper. | The Yang-Mills flow near the boundary of Teichmueller space | 14,215 |
A gauge theoretic description of the Morgan-Shalen compactification of the $\SL$ character variety of the fundamental group of a hyperbolic surface is given in terms of a natural compactification of the moduli space of Higgs bundles via the Hitchin map. | On the Morgan-Shalen compactification of the SL(2,C) character varieties
of surface groups | 14,216 |
In the framework of Abstract Differential Geometry, we show that to a given principal sheaf and a representation of its stuctural sheaf in $A^n$, where A is a sheaf of associative, commutative, unital algebras (over R or C), we associate a vector sheaf. Moreover, under some natural assumptions on the compatibility of the representation with the Maurer-Cartan (or, logarithmic) differentials of the structural sheaves involved, we also show that the connections of the principal sheaf induce A-connections of the associated vector sheaf. We thus recover a result of the classical Differential Geometry within a purely algebro-topological context, without any smoothness assumption. | Vector Sheaves Associated with Principal Sheaves | 14,217 |
A-manifolds and A-bundles are manifolds and vector bundles modelled on a projective finitely generated module over a topological algebra A. In this paper we investigate the conditions under which an A-bundle is provided with an A-valued hermitian structure and a compatible connection, in case A is a commutative complete locally m-convex C*-algebra with unit. In this investigation two obstacles appear: first, A-manifolds do not admit partitions of unity in the classical sence, and, secondly, the existence of a hermitian structure is not equivalent to the reduction of the structural group of the bundle to a certain subgroup. However, we prove that if the bundle has a trivializing covering whose transition functions take values in the group of the "A-hermitian product preserving" automorphisms of the fibre type and the base space admits at least one A-valued partition of unity (subordinate to this covering), then the A-bundle admits an A-hermitian structure and a compatible connection. | Hermitian Structures and Compatible Connections on A-bundles | 14,218 |
Let M be a closed connected manifold. Let m(M) be the Morse number of M, that is, the minimal number of critical points of a Morse function on M. Let N be a finite cover of M of degree d. M.Gromov posed the following question: what are the asymptotic properties of m(N) as d goes to infinity? In this paper we study the case of high dimensional manifolds M with free abelian fundamental group. Let x be a non-zero element of H^1(M), let M(x) be the infinite cyclic cover corresponding to x, and t be a generator of the structure group of this cover. Set M(x,k)=M(x)/t^k. We prove that the sequence m(M(x,k))/k converges as k goes to infinity. For x outside of a finite union of hyperplanes in H^1(M) we obtain the asymptotics of m(M(x,k)) as k goes to infinity, in terms of homotopy invariants of M related to Novikov homology of M. | On the asymptotics of Morse numbers of finite covers of manifolds | 14,219 |
Various transformations of isothermic surfaces are discussed and their interrelations are analyzed. Applications to cmc-1 surfaces in hyperbolic space and their minimal cousins in Euclidean space are presented: the Umehara-Yamada perturbation, the classical and Bryant's Weierstrass type representations, and the duality for cmc-1 surfaces are interpreted in terms of transformations of isothermic surfaces. A new Weierstrass type representation is introduced and a Moebius geometric characterization of cmc-1 surfaces in hyperbolic space and minimal surfaces in Euclidean space is given. | Moebius geometry of surfaces of constant mean curvature 1 in hyperbolic
space | 14,220 |
Suppose M_t is a smooth family of compact connected two dimensional submanifolds of Euclidean space E^3 without boundary varying isometrically in their induced Riemannian metrics. Then we show that the mean curvature integrals over M_t are constant. It is unknown whether there are nontrivial such bendings. The estimates also hold for periodic manifolds for which there are nontrivial bendings. In addition, our methods work essentially without change to show the similar results for submanifolds of H^n and S^n. The rigidity of the mean curvature integral can be used to show new rigidity results for isometric embeddings and provide new proofs of some well-known results. This, together with far-reaching extensions of the results of the present note is done in the preprint: I Rivin, J-M Schlenker, Schlafli formula and Einstein manifolds, IHES preprint (1998). Our result should be compared with the well-known formula of Herglotz. | The mean curvature integral is invariant under bending | 14,221 |
This paper is a survey of some of the developments in coarse extrinsic geometry since its inception in the work of Gromov. Distortion, as measured by comparing the diameter of balls relative to different metrics, can be regarded as one of the simplist extrinsic notions. Results and examples concerning distorted subgroups, especially in the context of hyperbolic groups and symmetric spaces, are exposed. Other topics considered are quasiconvexity of subgroups; behaviour at infinity, or more precisely continuous extensions of embedding maps to Gromov boundaries in the context of hyperbolic groups acting by isometries on hyperbolic metric spaces; and distortion as measured using various other filling invariants. | Coarse extrinsic geometry: a survey | 14,222 |
We extend the notion of basic classes (for the Donaldson invariants) to 4-manifolds with $b^+>1$ which are (potentially) not of simple type or satisfy $b_1 >0$. We also give a structure theorem for the Donaldson invariants of 4-manifolds with $b^+>1$, $b_1>0$ and of strong simple type. | Basic classes for four-manifolds not of simple type | 14,223 |
A smooth four manifold is of finite type $r$ if its Donaldson invariant satisfies D((x^2-4)^r)=0. We prove that every simply connected manifold is of finite type by using the structure of Donaldson invariants in the presence of immersed spheres. More precisely we prove that if a manifold X contains an immersed sphere with $p$ positive double points and a non-negative self-intersection $a$, then it is of finite type with r = [(2p+2-a)/4]. | Immersed spheres and finite type of Donaldson invariants | 14,224 |
Lorentzian versions of classical Riemannian volume comparison theorems by Gunther, Bishop and Bishop-Gromov, are stated for suitable natural subsets of general semi-Riemannian manifolds. The problem is more subtle in the Bishop-Gromov case, which is extensively discussed. For the general semi-Riemannian case, a local version of the Gunther and Bishop theorems is given and applied. | Some semi-Riemannian volume comparison theorems | 14,225 |
This paper demonstrates the power of the calculus developed in the two previous parts of the series for all real forms of the almost Hermitian symmetric structures on smooth manifolds, including e.g. conformal Riemannian and almost quaternionic geometries. Exploiting some finite dimensional representation theory of simple Lie algebras, we give explicit formulae for distinguished invariant curved analogues of the standard operators in terms of the linear connections belonging to the structures in question, so in particular we prove their existence. Moreover, we prove that these formulae for k-th order standard operators, k=1,2,..., are universal for all geometries in question. | Invariant operators on manifolds with almost Hermitian symmetric
structures, III. Standard operators | 14,226 |
We compare the eigenvalues of the Dirac and Laplace operator on a two-dimensional torus with respect to the trivial spin structure. In particular, we compute their variation up to order 4 upon deformation of the flat metric, study the corresponding Hamiltonian and discuss several families of examples. | A comparison of the eigenvalues of the Dirac and Laplace operator on the
two-dimensional torus | 14,227 |
We prove necessary and sufficient conditions for a smooth surface in a 4-manifold X to be pseudoholomorphic with respect to some almost complex structure on X. This provides a systematic approach to the construction of pseudoholomorphic curves that do not minimize the genus in their homology class. | Embedded surfaces and almost complex structures | 14,228 |
Representations of Dirac-Hestenes and Dirac spinor fields via coordinates of surfaces conformally immersed into 4-dimensional complex space are proposed. A relation between time evolution of spinor fields and integrable deformations of surfaces is discussed. | Dirac-Hestenes spinors and Weierstrass representation for surfaces in 4D
complex space | 14,229 |
One of the basic objects in the Morse theory of circle-valued maps is Novikov complex - an analog of the Morse complex of Morse functions. Novikov complex is defined over the ring of Laurent power series with finite negative part. The main aim of this paper is to present a detailed and self-contained exposition of the author's theorem saying that C^0-generically the Novikov complex is defined over the ring of rational functions. The paper contains also a systematic treatment of the topics of the classical Morse theory related to Morse complexes (Chapter 2). We work with a new class of gradient-type vector fields, which includes riemannian gradients. In Ch.3 we suggest a purely Morse-theoretic (not using triangulations) construction of small handle decomposition of manifolds. In the Ch.4 we deal with the gradients of Morse functions on cobordisms. Due to the presence of critical points the descent along the trajectories of such gradient does no define in general a continuous map from the upper component of the boundary to the lower one. We show that for C^0-generic gradients there is an algebraic model of "gradient descent map". This is one of the main tools in the proof of the main theorem (Chapter 5). We give also the generalizations of the result for the versions of Novikov complex defined over completions of group rings (non commutative in general). | C^0-generic properties of boundary operators in Novikov Complex | 14,230 |
We prove new adjunction inequalities for embedded surfaces in four-manifolds with non-negative self-intersection number by using the Donaldson invariants. These formulas are completely analogous to the ones obtained by Ozsv\'ath and Szab\'o using the Seiberg-Witten invariants. To prove these relations, we give a fairly explicit description of the structure of the Fukaya-Floer homology of a surface times a circle. As an aside, we also relate the Floer homology of a surface times a circle with the cohomology of some symmetric products of the surface. | Higher type adjunction inequalities for Donaldson invariants | 14,231 |
We prove a formula relating the analytic torsion and Reidemeister torsion on manifolds with boundary in the general case when the metric is not necessarily a product near the boundary. The product case has been established by W. Lu\"ck and S. M. Vishik. We find that the extra term that comes in here in the nonproduct case is the transgression of the Euler class in the even dimensional case and a slightly more mysterious term involving the second fundamental form of the boundary and the curvature tensor of the manifold in the odd dimensional case. | Analytic Torsion and R-Torsion for Manifolds with Boundary | 14,232 |
We describe an effective algorithm for computing Seiberg-Witen invariants of lens spaces. We apply it to two problems: (i) to compute the Froyshov invariants of a large family of lens spaces; (ii) to show that the knowledge of the Seiberg-Witten invariants of lens spaces is topologically equivalent to the knowledge of its Casson-Walker invariant and of its Milnor-Turaev torsion. We also use the results in problem (i) to derive several topological results concerning the negative definite manifolds bounding a given lens space. | Seiberg-Witten theoretic invariants of lens spaces | 14,233 |
We prove the vanishing of the Dolbeault cohomology groups on Hermitian manifolds with $dd^c$-harmonic K\"ahler form and positive (1,1)-part of the Ricci form of the Bismut connection. This implies the vanishing of the Dolbeault cohomology groups on complex surfaces which admit a conformal class of Hermitian metrics, such that the Ricci tensor of the canonical Weyl structure is positive. As a corollary we obtain that any such surface must be rational with $c_1^2 >0$. As an application, the pth Dolbeault cohomology groups of a left-invariant complex structure compatible with a bi-invariant metric on a compact even dimensional Lie group are computed. | Vanishing theorems on Hermitian manifolds | 14,234 |
In this paper, the Dirac, twistor and Killing equations on Weyl manifolds with CSpin structures are investigated. A conformal Schr"odinger-Lichnerowicz formula is presented and used to show integrability conditions for these equations. By introducing the Killing equation for spinors of arbitrary weight, the result of Andrei Moroianu in [9] is generalized in the following sense. The only non-closed Weyl manifolds of dimension greater than 3 that admit solutions of the real Killing equation are 4-dimensional and non-compact. Any Weyl manifold of these dimensions admitting a real Killing spinor has to be Einstein-Weyl. | Spinor equations in Weyl geometry | 14,235 |
The energy minimization problem associated to uniform, isotropic, linearly elastic rods leads to a geometric variational problem for the rod centerline, whose solutions include closed, knotted curves. We give a complete description of the space of closed and quasiperiodic solutions. The quasiperiodic curves are parametrized by a two-dimensional disc. The closed curves arise as a countable collection of one-parameter families, connecting the m-fold covered circle to the n-fold covered circle for any m,n relatively prime. Each family contains exactly one self-intersecting curve, one elastic curve, and one closed curve of constant torsion. Two torus knot types are represented in each family, and all torus knots are represented by elastic rod centerlines. | Knot types, homotopies and stability of closed elastic rods | 14,236 |
We endow the group of invertible Fourier integral operators on an open}manifold with the structure of an ILH Lie group. This is done by establishing such structures for the groups of invertible pseudodifferential operators and contact transformations on an open manifold of bounded geometry, and gluing those together via a local section. | Lie Groups of Fourier Integral Operators on Open Manifolds | 14,237 |
It is shown that every bundle $\varSigma\to M$ of complex spinor modules over the Clifford bundle $\Cl(g)$ of a Riemannian space $(M,g)$ with local model $(V,h)$ is associated with an lpin ("Lipschitz") structure on $M$, this being a reduction of the ${\Ort}(h)$-bundle of all orthonormal frames on M to the Lipschitz group $\Lpin(h)$ of all automorphisms of a suitably defined spin space. An explicit construction is given of the total space of the $\Lpin(h)$-bundle defining such a structure. If the dimension m of M is even, then the Lipschitz group coincides with the complex Clifford group and the lpin structure can be reduced to a pin$^{c}$ structure. If m=2n-1, then a spinor module $\varSigma$ on M is of the Cartan type: its fibres are 2^n-dimensional and decomposable at every point of M, but the homomorphism of bundles of algebras $\Cl(g)\to\End\varSigma$ globally decomposes if, and only if, M is orientable. Examples of such bundles are given. The topological condition for the existence of an lpin structure on an odd-dimensional Riemannian manifold is derived and illustrated by the example of a manifold admitting such a structure, but no pin^c structure. | Spin spaces, Lipschitz groups, and spinor bundles | 14,238 |
A subset S of a Riemannian manifold N is called extrinsically homogeneous if S is an orbit of a subgroup of the isometry group of N. Thorbergsson proved the remarkable result that every complete, connected, full, irreducible isoparametric submanifold of a finite dimensional Euclidean space of rank at least 3 is extrinsically homogeneous. This result, combined with results of Palais-Terng and Dadok, finally classified irreducible isoparametric submanifolds of a finite dimensional Euclidean space of rank at least 3. While Thorbergsson's proof used Tits buildings, a simpler proof without using Tits buildings was given by Olmos. The main purpose of this paper is to extend Thorbergsson's result to the infinite dimensional case. | Homogeneity of infinite dimensional isoparametric submanifolds | 14,239 |
We consider the Dolbeault operator of $K^{1/2}$ -- the square root of the canonical line bundle which determines the spin structure of a compact Hermitian spin surface (M,g,J). We prove that the Dolbeault cohomology groups of $K^{1/2}$ vanish if the scalar curvature of g is non-negative and non-identically zero. Moreover, we estimate the first eigenvalue of the Dolbeault operator when the conformal scalar curvature k is non-negative and when k is positive. In the first case we give a complete list of limiting manifolds and in the second one we give non-K\"ahler examples of limiting manifolds. | The Dolbeault operator on Hermitian spin surfaces | 14,240 |
We prove the vanishing of the first Betti number on compact manifolds admitting a Weyl structure whose Ricci tensor satisfies certain positivity conditions, thus obtaining a Bochner-type vanishing theorem in Weyl geometry. We also study compact Hermitian-Weyl manifolds with non-negative symmetric part of the Ricci tensor of the canonical Weyl connection and show that every such manifold has first Betti number $b_1 =1$ and Hodge numbers $h^{p,0} =0$ for $p>0$, $h^{0,1} =1$, $h^{0,q} =0$ for $q>1$. | Weyl structures with positive Ricci tensor | 14,241 |
The authors study the geometry of lightlike hypersurfaces on manifolds $(M, c)$ endowed with a pseudoconformal structure $c = CO (n - 1, 1)$ of Lorentzian signature. Such hypersurfaces are of interest in general relativity since they can be models of different types of physical horizons. On a lightlike hypersurface, the authors consider the fibration of isotropic geodesics and investigate their singular points and singular submanifolds. They construct a conformally invariant normalization of a lightlike hypersurface intrinsically connected with its geometry and investigate affine connections induced by this normalization. The authors also consider special classes of lightlike hypersurfaces. In particular, they investigate lightlike hypersurfaces for which the elements of the constructed normalization are integrable. | Lightlike hypersurfaces on manifolds endowed with a conformal structure
of Lorentzian signature | 14,242 |
The authors study the geometry of lightlike hypersurfaces on pseudo-Riemannian manifolds $(M, g)$ of Lorentzian signature. Such hypersurfaces are of interest in general relativity since they can be models of different types of physical horizons. For a lightlike hypersurface $V \subset (M, g)$ of general type and for some special lightlike hypersurfaces (namely, for totally umbilical and belonging to a manifold $(M, g)$ of constant curvature), in a third-order neighborhood of a point $x \in V$, the authors construct invariant normalizations intrinsically connected with the geometry of $V$ and investigate affine connections induced by these normalizations. For this construction, they used relative and absolute invariants defined by the first and second fundamental forms of $V$. The authors show that if $\dim M = 4$, their methods allow to construct three invariant normalizations and affine connections intrinsically connected with the geometry of $V$. Such a construction is given in the present paper for the first time. The authors also consider the fibration of isotropic geodesics of $V$ and investigate their singular points and singular submanifolds. | On some methods of construction of invariant normalizations of lightlike
hypersurfaces | 14,243 |
I point out some very elementary examples of special Lagrangian tori in certain Calabi-Yau manifolds that occur as hypersurfaces in complex projective space. All of these are constructed as real slices of smooth hypersurfaces defined over the reals. This method of constructing special Lagrangian submanifolds is well known. What does not appear to be in the current literature is an explicit description of such examples in which the special Lagrangian submanifold is a 3-torus. | Some examples of special Lagrangian tori | 14,244 |
We investigate the twistor space and the Grassmannian fibre bundle of a Lorentzian 4-space with natural almost optical structures and its induced CR-structures. The twistor spaces of the Lorentzian space forms $\R^4_1, \Di{S}^4_1$ and $\Di{H}^4_1$ are explicitly discussed. The given twistor construction is applied to surface theory in Lorentzian 4-spaces. Immersed spacelike surfaces in a Lorentzian 4-space with special geometric properties like semi-umbilic surfaces and surfaces that have mean curvature of vanishing lengths correspond to holomorphic curves in the Lorentzian twistor space. For the Lorentzian space forms $\R^4_1, \Di{S}^4_1$ and $\Di{H}^4_1$ those surfaces are explicitly constructed and classified. | Twistorial constructions of spacelike surfaces in Lorentzian 4-manifolds | 14,245 |
We analyze the limit of the p-form Laplacian under a collapse, with bounded sectional curvature and bounded diameter, to a smooth limit space. As an application, we characterize when the p-form Laplacian has small positive eigenvalues in a collapsing sequence. | Collapsing and the Differential Form Laplacian : The Case of a Smooth
Limit Space | 14,246 |
The authors study the geometry of lightlike hypersurfaces on a four-dimensional manifold $(M, c)$ endowed with a pseudoconformal structure $c = CO (2, 2)$. They prove that a lightlike hypersurface $V \subset (M, c)$ bears a foliation formed by conformally invariant isotropic geodesics and two isotropic distributions tangent to these geodesics, and that these two distributions are integrable if and only if $V$ is totally umbilical. The authors also indicate how, using singular points and singular submanifolds of a lightlike hypersurface $V \subset (M, c)$, to construct an invariant normalization of $V$ intrinsically connected with $V$. | Lightlike hypersurfaces on a four-dimensional manifold endowed with a
pseudoconformal structure of signature (2, 2) | 14,247 |
We derive a decomposition formula for the spectral flow of a 1-parameter family of self-adjoint Dirac operators on an odd-dimensional manifold $M$ split along a hypersurface $\Sigma$ ($M=X\cup_{\Sigma} Y$). No transversality or stretching hypotheses are assumed and the boundary conditions can be chosen arbitrarily. The formula takes the form $SF(D)= SF(D_{|X}, B_X) + SF(D_{|Y},B_Y) + \mu(B_Y,B_X) + S$ where $B_X$ and $B_Y$ are boundary conditions, $\mu$ denotes the Maslov index, and $S$ is a sum of explicitly defined Maslov indices coming from stretching and rotating boundary conditions. The derivation is a simple consequence of Nicolaescu's theorems and elementary properties of the Maslov index. We show how to use the formula and derive many of the splitting theorems in the literature as simple consequences. | A general splitting formula for the spectral flow | 14,248 |
If $p : Y \to X$ is an unramified covering map between two compact oriented surfaces of genus at least two, then it is proved that the embedding map, corresponding to $p$, from the Teichm\"uller space ${\cal T}(X)$, for $X$, to ${\cal T}(Y)$ actually extends to an embedding between the Thurston compactification of the two Teichm\"uller spaces. Using this result, an inductive limit of Thurston compactified Teichm\"uller spaces has been constructed, where the index for the inductive limit runs over all possible finite unramified coverings of a fixed compact oriented surface of genus at least two. This inductive limit contains the inductive limit of Teichm\"uller spaces, constructed in \cite{BNS}, as a subset. The universal commensurability modular group, which was constructed in \cite{BNS}, has a natural action on the inductive limit of Teichm\"uller spaces. It is proved here that this action of the universal commensurability modular group extends continuously to the inductive limit of Thurston compactified Teichm\"uller spaces. | Thurston boundary of Teichmüller spaces and the commensurability
modular group | 14,249 |
For a closed K\"{a}hler manifold with a Hamiltonian action of a connected compact Lie group by holomorphic isometries, we construct a formal Frobenius manifold structure on the equivariant cohomology by exploiting a natural DGBV algebra structure on the Cartan model. | Formal Frobenius manifold structure on equivariant cohomology | 14,250 |
The one-skeleton of a G-manifold M is the set of points p in M where $\dim G_p \geq \dim G -1$; and M is a GKM manifold if the dimension of this one-skeleton is 2. Goresky, Kottwitz and MacPherson show that for such a manifold this one-skeleton has the structure of a ``labeled" graph, $(\Gamma, \alpha)$, and that the equivariant cohomology ring of M is isomorphic to the ``cohomology ring'' of this graph. Hence, if M is symplectic, one can show that this ring is a free module over the symmetric algebra $\SS(\fg^*)$, with $b_{2i}(\Gamma)$ generators in dimension 2i, $b_{2i}(\Gamma)$ being the ``combinatorial'' 2i-th Betti number of $\Gamma$. In this article we show that this ``topological'' result is , in fact, a combinatorial result about graphs. | One-skeleta, Betti numbers and equivariant cohomology | 14,251 |
Let G be a connected Lie group with Lie algebra g. The Duflo map is a vector space isomorphism between the symmetric algebra S(g) and the universal enveloping algebra U(g) which, as proved by Duflo, restricts to a ring isomorphism from invariant polynomials onto the center of the universal enveloping algebra. The Duflo map extends to a linear map from compactly supported distributions on the Lie algebra g to compactly supported distributions on the Lie group G, which is a ring homomorphism for G-invariant distributions. In this paper we obtain analogues of the Duflo map and of Duflo's theorem in the context of equivariant cohomology of G-manifolds. Our result involves a non-commutative version of the Weil algebra and of the de Rham model of equivariant cohomology. | The non-commutative Weil algebra | 14,252 |
Three great theorems of Thurston read: Haken manifolds are hyperbolic; big ramified coverings are hyperbolic; big surgeries are hyperbolic. Recent developments indicate that the later two theorems are essentially a corollary of the first, that is there are much more Haken manifolds than expected by Thurston. In fact Freedman showed very recently that big ramified coverings are Haken. A version of his proof with various improvements was obtained by Cooper-Long and Cooper-Long-Reid. The two main contributions of the present paper are the following. I give an analytic proof of Freedman's result and the improved version of Cooper-Long-Reid. This proof is based on fundamentally different approach than Freedman's and is 10 times shorter, but uses the full forse of the hyperbolization theorem. Secondly, I prove that ANY ramified covering of a tight knot is Haken. Morover any knot becomes tight after a big surgery. So any ramified covering of a big surgery is Haken. Many other results of the paper are better seen from the introduction. The paper uses many different techniques and may be difficult to read for a beginner. | Hakenness and b_1 | 14,253 |
We use Floer's exact triangle to study the u-map (cup product with the 4-dimensional class) in the Floer cohomology groups of admissible SO(3) bundles over closed, oriented 3-manifolds. In the case of non-trivial bundles we show that (u^2-64)^n = 0 for some positive integer n. For homology 3-spheres Y the same holds for a certain reduced Floer group, which is obtained from the ordinary one by factoring out interaction with the trivial connection. This leads to a new proof (in the simply-connected case) of the finite type conjecture of Kronheimer and Mrowka concerning the structure of Donaldson polynomials. In the case of rational coefficients, interaction with the trivial connection is measured by a single integer h(Y), which is additive under connected sums and depends only on the rational homology cobordism class of Y. | Equivariant aspects of Yang-Mills Floer theory | 14,254 |
In this paper, we prove results concerning the large scale geometry of connected, simply connected nonabelian nilpotent Lie groups equipped with left invariant Riemannian metrics. Precisely, we prove that there do not exist quasi-isometric embeddings of such a nilpotent Lie group into either a CAT(0) metric space or an Alexandrov metric space with curvature bounded below. The main technical aspect of this work is the proof of a limited metric differentiability of Lipschitz maps between connected graded nilpotent Lie groups equipped with left invariant Carnot-Caratheodory metrics and complete metric spaces. | The large scale geometry of nilpotent Lie groups | 14,255 |
Some of the most important classes of surfaces in projective 3-space are reviewed: these are isothermally asymptotic surfaces, projectively applicable surfaces, surfaces of Jonas, projectively minimal surfaces, etc. It is demonstrated that the corresponding projective "Gauss-Codazzi" equations reduce to integrable systems which are quite familiar from the modern soliton theory and coincide with the stationary flows in the Davey-Stewartson and Kadomtsev-Petviashvili hierarchies, equations of the Toda lattice, etc. The corresponding Lax pairs can be obtained by inserting a spectral parameter in the equations of the Wilczynski moving frame. | Integrable systems in projective differential geometry | 14,256 |
We consider complete noncompact Riemannian manifolds with quadratically decaying lower Ricci curvature bounds and minimal volume growth. We first prove a rigidity result showing that ends with strongly minimal volume growth are isometric to warped product manifolds. Next we consider the almost rigid case in which manifolds with nonnegative and quadratically decaying lower Ricci curvature bounds have minimal volume growth. Compact regions in such manifolds are shown to be asymptotically close to warped products in the Gromov-Hausdorff topology. Manifolds with nonnegative Ricci curvature and linear volume growth are shown to have regions which are asymptotically close to being isometric products. The proofs involve a careful analysis of the Busemann functions on these manifolds using the recently developed Cheeger-Colding Almost Rigidity Theory. In addition, we show that the diameters of the level sets of Busemann functions in such manifolds grow sublinearly. | The Almost Rigidity of Manifolds with Lower Bounds on Ricci Curvature
and Minimal Volume Growth | 14,257 |
Lower bounds on Ricci curvature limit the volumes of sets and the existence of harmonic functions on Riemannian manifolds. In 1975, Shing Tung Yau proved that a complete noncompact manifold with nonnegative Ricci curvature has no nonconstant harmonic functions of sublinear growth. In the same paper, Yau used this result to prove that a complete noncompact manifold with nonnegative Ricci curvature has at least linear volume growth. In this paper, we prove the following theorem concerning harmonic functions on these manifolds. Theorem: Let M be a complete noncompact manifold with nonnegative Ricci curvature and at most linear volume growth. If there exists a nonconstant harmonic function, f, of polynomial growth of any given degree q, then the manifold splits isometrically, M= N x R. | Harmonic Functions on Manifolds with Nonnegative Ricci Curvature and
Linear Volume Growth | 14,258 |
This is primarily an expository note showing that earlier work of Lai on CR geometry provides a clean interpretation, in terms of a Gauss map, for an adjunction formula for embedded surfaces in an almost complex four manifold. We will see that if F is a surface with genus g in an almost complex four-manifold M, then 2 - 2 g + F \cdot F - i^{*} c_{1}(M) - 2 F\cdot C = 0, where C is a two-cycle on M pulled back from the cycle of two planes with complex structure in a Grassmannian Gr (2, C^N) via a Gauss map and where i^{*} c_{1}(M) is the restriction of the first Chern class of M to F. The key new term of interest is F \cdot C, which will capture the points of F whose tangent planes inherit a complex structure from the almost complex structure of the ambient manifold M. These complex jump points then determine the genus of smooth representative of a homology class in H_{2}(M, Z). Further, via polarization, we can use this formula to determine the intersection form on M from knowing the nature of the complex jump points of M's surfaces. | Intersection Forms and the Adjunction Formula for Four-manifolds via CR
Geometry | 14,259 |
This paper concerns complete noncompact manifolds with nonnegative Ricci curvature. Roughly, we say that M has the loops to infinity property if given any noncontractible closed curve, C, and given any compact set, K, there exists a closed curve contained in M\K which is homotopic to C. The main theorems in this paper are the following. Theorem I: If M has positive Ricci curvature then it has the loops to infinity property. Theorem II: If M has nonnegative Ricci curvature then it either has the loops to infinity property or it is isometric to a flat normal bundle over a compact totally geodesic submanifold and its double cover is split. Theorem III: Let M be a complete riemannian manifold with the loops to infinity property along some ray starting at a point, p. Let D containing p be a precompact region with smooth boundary and S be any connected component of the boundary containing a point, q, on the ray. Then the map from the fundamental group of S based at q to the fundamental group of Cl(D) based at p induced by the inclusion map is onto. | On Loops Representing Elements of the Fundamental Group of a Complete
Manifold with Nonnegative Ricci Curvature | 14,260 |
We develop a geometric scattering theory for a geometrically finite group acting on (a vector bundle over) a symmetric space of negative curvature. In particular, we obtain the meromorphic continuation of Eisenstein series and scattering matrices and their functional equations. | Scattering theory for geometrically finite groups | 14,261 |
We continue the investigation of the correspondence between systems of conservation laws and congruences of lines in projective space. Relationship between "additional" conservation laws and hypersurfaces conjugate to a congruence is established. This construction allows us to introduce, in a purely geometric way, the L\'evy transformations of semihamiltonian systems. Correspondence between commuting flows and certain families of planes containing the lines of the congruence is pointed out. In the particular case n=2 this construction provides an explicit parametrization of surfaces, harmonic to a given congruence. Adjoint L\'evy transformations of semihamiltonian systems are discussed. Explicit formulae for the L\'evy and adjoint L\'evy transformations of the characteristic velocities are set down. A closely related construction of the Ribaucour congruences of spheres is discussed in the Appendix. | Transformations of quasilinear systems originating from the projective
theory of congruences | 14,262 |
We study compact stable embedded minimal surfaces whose boundary is given by two collections of closed smooth Jordan curves in close planes of Euclidean 3-space. Our main result is a classification of these minimal surfaces, under certain natural geometric asymptotic constraints, in terms of certain associated varifolds which can be enumerated explicitely. One consequence of this result is the uniqueness of the area minimizing examples. Another is the asymptotic nonexistence of stable compact embedded minimal surfaces of positive genus bounded by two convex curves in parallel planes. | Classification of Stable Minimal Surfaces Bounded by Jordan Curves in
Close Planes | 14,263 |
We determine the Seiberg-Witten-Floer homology groups of the three-manifold which is the product of a surface of genus $g \geq 1$ times the circle, together with its ring structure, for spin-c structures which are non-trivial on the three-manifold. We give applications to computing Seiberg-Witten invariants of four-manifolds which are connected sums along surfaces and also we reprove the higher type adjunction inequalities previously obtained by Oszv\'ath and Szab\'o. | Seiberg-Witten-Floer homology of a surface times a circle for
non-torsion spin-c structures | 14,264 |
Let $M$ be a smooth manifold, $\cal S$ the space of polynomial on fibers functions on $T^*M$ (i.e., of symmetric contravariant tensor fields). We compute the first cohomology space of the Lie algebra, $Vect(M)$, of vector fields on $M$ with coefficients in the space of linear differential operators on $\cal S$. This cohomology space is closely related to the $Vect(M)$-modules, ${\cal D}_\lambda(M)$, of linear differential operators on the space of tensor densities on $M$ of degree $\lambda$. | Cohomology of the vector fields Lie algebra and modules of differential
operators on a smooth manifold | 14,265 |
In this note, we look at estimates for the scalar curvature k of a Riemannian manifold M which are related to spin^c Dirac operators: We show that one may not enlarge a Kaehler metric with positive Ricci curvature without making k smaller somewhere on M. We also give explicit upper bounds for min(k) for arbitrary Riemannian metrics on certain submanifolds of complex projective space. In certain cases, these estimates are sharp: we give examples where equality is obtained. | Spin^c Structures and Scalar Curvature Estimates | 14,266 |
We construct exact solutions of the Einstein-Dirac equation, which couples the gravitational field with an eigenspinor of the Dirac operator via the energy-momentum tensor. For this purpose we introduce a new field equation generalizing the notion of Killing spinors. The solutions of this spinorial field equation are called weak Killing spinors (WK-spinors). They are special solutions of the Einstein-Dirac equation and in dimension n=3 the two equations essentially coincide. It turns out that any Sasakian manifold with Ricci tensor related in some special way to the metric tensor as well as to the contact structure admits a WK-spinor. This result is a consequence of the investigation of special spinorial field equations on Sasakian manifolds (Sasakian quasi-Killing spinors). Altogether, in odd dimensions a contact geometry generates a solution of the Einstein-Dirac equation. Moreover, we prove the existence of solutions of the Einstein-Dirac equations that are not WK-spinors in all dimensions n > 8. | The Einstein-Dirac Equation on Riemannian Spin Manifolds | 14,267 |
We begin a systematic study of these spaces, initially following along the lines of Eberlein's comprehensive study of the Riemannian case. In particular, we integrate the geodesic equation, discuss the structure of the isometry group, and make a study of lattices and periodic geodesics. Some major differences from the Riemannian theory appear. There are many flat groups (versus none), including Heisenberg groups. While still a semidirect product, the isometry group can be strictly larger than the obvious analogue. Everything is illustrated with explicit examples. We introduce the notion of pH-type, which refines Kaplan's H-type and completes Ciatti's partial extension. We give a general construction for algebras of pH-type. | Pseudoriemannian 2-Step Nilpotent Lie Groups | 14,268 |
We introduce the self-linking number of a smooth closed curve in R^n with respect to a 3-dimensional vector bundle over the curve, provided that some regularity conditions are satisfied. When n=3, this construction gives the classical self-linking number of a closed embedded curve with non-vanishing curvature. We also look at some interesting particular cases, which correspond to the osculating or the orthogonal vector bundle of the curve. | The self-linking number of a closed curve in R^n | 14,269 |
Sei G eine zusammenh\"angende reduktive komplexe algebraische Gruppe, die auf einer glatten affinen komplexen Variet\"at M wirke, und bezeichne $\Diff[G]{M}$ die G-invarianten algebraischen Differentialoperatoren auf M. Zerlegt man den Koordinatenring $\Aff{\C}{M}$ in G-isotypische Komponenten, so zeigen wir, da{\ss} die hierbei auftretenden Vielfachheitenr\"aume irreduzible $\Diff[G]{M}$-Moduln sind, zentralen Charakter haben und durch diesen eindeutig bestimmt sind. Anschlie{\ss}end beschreiben wir die analoge Zerlegung f\"ur reelle Formen und zeigen anhand einiger singul\"arer Beispiele, da{\ss} f\"ur nicht glatte Variet\"aten \"ahnliche Ergebnisse nicht zu erwarten sind. | Invariante Differentialoperatoren und die Frobenius-Zerlegung einer
G-Variet"at | 14,270 |
Jeffrey and Kirwan suggested expressions for intersection pairings on the reduced space of a Hamiltonian G-space in terms of multiple residues. In this paper we prove a residue formula for symplectic volumes of reduced spaces of a quasi-Hamiltonian SU(2)-space. The definition of quasi-Hamiltonian G-spaces was recently introduced by Alekseev, Malkin and Meinrenken. | A Residue Formula for SU(2)-valued Moment Maps | 14,271 |
Spinor fields on surfaces of revolution conformally immersed into 3-dimensional space are considered in the framework of the spinor representations of surfaces. It is shown that a linear problem (a 2-dimensional Dirac equation) related with a modified Veselov- Novikov hierarchy in the case of the surface of revolution reduces to a well-known Zakharov-Shabat system. In the case of one-soliton solution an explicit form of the spinor fields is given by means of linear Bargmann potentials and is expressed via the Jost functions of the Zakharov-Shabat system. It is shown also that integrable deformations of the spinor fields on the surface of revolution are defined by a modified Korteweg-de Vries hierarchy. | Spinor Fields on the Surface of Revolution and their Integrable
Deformations via the mKdV-Hierarchy | 14,272 |
We generalize the well-known lower estimates for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold proved by Th. Friedrich (1980) and O. Hijazi (1986, 1992). The special solutions of the Einstein-Dirac equation constructed recently by Friedrich/Kim are examples for the limiting case of these inequalities. The discussion of the limiting case of these estimates yields two new field equations generalizing the Killing equation as well as the weak Killing equation for spinor fields. Finally, we discuss the 2- and 3-dimensional case in more detail. | Some Remarks on the Hijazi Inequality and Generalizations of the Killing
Equation for Spinors | 14,273 |
In this work we analyze the behavior of Massey products of closed manifolds under the blow-up construction. The results obtained in the article are applied to the problem of constructing closed symplectic non-formal manifolds. The proofs use Thom spaces as an important technical tool. This application of Thom spaces is of conceptual interest. | On Thom spaces, Massey products and non-formal symplectic manifolds | 14,274 |
We describe a family of locally conformal Kaehler metrics on class 1 Hopf surfaces H containing some recent metrics constructed by P. Gauduchon and L. ornea. We study some canonical foliations associated to these metrics, in particular a 2-dimensional foliation E that is shown to be independent of the metric. We elementary prove that E has compact leaves if and only if H is elliptic. In this case the leaves of E give explicitly the elliptic fibration of H, and the natural orbifold structure on the leaf space is illustrated. | Hopf surfaces: a family of locally conformal Kaehler metrics and
elliptic fibrations | 14,275 |
The subgroups of GL(n,R) that act irreducibly on R^n and that can occur as the holonomy of a torsion-free affine connection on an n-manifold are classified, thus completing the work on this subject begun by M. Berger in the 1950s. The methods employed include representation theory, the theory of hermitian symmetric spaces, twistor theory, and Poisson geometry. The latter theory is especially important for the construction and classification of those torsion-free connections whose holonomy falls into one of the so-called `exotic' cases, i.e., those that were not included in Berger's original lists. Some remarks involving an interpretation of some of the examples in terms of supersymmetric constructions are also included. | Classification of irreducible holonomies of torsion-free affine
connections | 14,276 |
We consider closed symplectically aspherical manifolds, i.e. closed symplectic manifolds $(M,\omega)$ satisfying the condition $[\omega]|_{\pi_2M}=0$. Rudyak and Oprea [RO] remarked that such manifolds have nice and controllable homotopy properties. Now it is clear that these properties are mostly determined by the fact that the strict category weight of $[\omega]$ equals 2. We apply the theory of strict category weight to the problem of estimating the number of closed orbits of charged particles in symplectic magnetic fields. In case of symplectically aspherical manifolds our theory enables us to improve some known estimations. | On symplectic manifolds with aspherical symplectic form | 14,277 |
We consider the flows generated by generic gradients of Morse maps of a closed connected manifold $M$ to a circle. To each such flow we associate an invariant counting the closed orbits of the flow. Each closed orbit is counted with the weight derived from its index and homotopy class. The resulting invariant is called the eta function, and lies in a suitable quotient of the Novikov completion of the group ring of the fundamental group of $M$. Its abelianization coincides with the logarithm of the twisted Lefschetz zeta function of the flow. For $C^0$-generic gradients we obtain a formula expressing the eta function in terms of the torsion of a special homotopy equivalence between the Novikov complex of the gradient flow and the completed simplicial chain complex of the universal cover of $M$. | Closed orbits of gradient flows and logarithms of non-abelian Witt
vectors | 14,278 |
Let $S$ be a set of critical points of a smooth real-valued function on a closed manifold $M$. Generalizing a well-known result of Lusternik--Schnirelmann, Reeken~[R] proved that $\cat S \geq \cat M$. Here we prove a generalization of Reeken"s inequality for gradient-like flows on compact spaces. | A remark on fixed point sets of gradient-like flows | 14,279 |
A flat complex vector bundle (E,D) on a compact Riemannian manifold (X,g) is stable (resp. polystable) in the sense of Corlette [C] if it has no D-invariant subbundle (resp. if it is the D-invariant direct sum of stable subbundles). It has been shown in [C] that the polystability of (E,D) in this sense is equivalent to the existence of a so-called harmonic metric in E. In this paper we consider flat complex vector bundles on compact Hermitian manifolds (X,g). We propose new notions of g-(poly-)stability of such bundles, and of g-Einstein metrics in them; these notions coincide with (poly-)stability and harmonicity in the sense of Corlette if g is a K\"ahler metric, but are different in general. Our main result is that the g-polystability in our sense is equivalent to the existence of a g-Hermitian-Einstein metric. Our notion of a g-Einstein metric in a flat bundle is motivated by a correspondence between flat bundles and Higgs bundles over compact surfaces, analogous to the correspondence in the case of K\"ahler manifolds [S1], [S2], [S3]. 1991 Mathematics Subject Classification: 53C07 | Flat connections, Higgs operators, and Einstein metrics on compact
Hermitian manifolds | 14,280 |
Let V be the pseudo-Euclidean vector space of signature (p,q), p>2 and W a module over the even Clifford algebra Cl^0 (V). A homogeneous quaternionic manifold (M,Q) is constructed for any spin(V)-equivariant linear map \Pi : \wedge^2 W \to V. If the skew symmetric vector valued bilinear form \Pi is nondegenerate then (M,Q) is endowed with a canonical pseudo-Riemannian metric g such that (M,Q,g) is a homogeneous quaternionic pseudo-K\"ahler manifold. The construction is shown to have a natural mirror in the category of supermanifolds. In fact, for any spin(V)-equivariant linear map \Pi : Sym^2 W \to V a homogeneous quaternionic supermanifold (M,Q) is constructed and, moreover, a homogeneous quaternionic pseudo-K\"ahler supermanifold (M,Q,g) if the symmetric vector valued bilinear form \Pi is nondegenerate. | A new construction of homogeneous quaternionic manifolds and related
geometric structures | 14,281 |
We construct new homogeneous Einstein spaces with negative Ricci curvature in two ways: First, we give a method for classifying and constructing a class of rank one Einstein solvmanifolds whose derived algebras are two-step nilpotent. As an application, we describe an explicit continuous family of ten-dimensional Einstein manifolds with a two-dimensional parameter space, including a continuous subfamily of manifolds with negative sectional curvature. Secondly, we obtain new examples of non-symmetric Einstein solvmanifolds by modifying the algebraic structure of non-compact irreducible symmetric spaces of rank greater than one, preserving the (constant) Ricci curvature. | New homogeneous Einstein metrics of negative Ricci curvature | 14,282 |
The approaches to quantum field theories based in the so called loop representation deserved much attention recently. In it, closed curves and holonomies around them play a central role. In this framework the group of loops and the group of hoops have been defined, the first one consisting in closed curves quotient with the equivalence relation that identifies curves differing in retraced segments, and the second one consisting in closed curves quotient with the equivalence relation that identifies curves having the same holonomy for every connection in a fiber bundle. The purpose of this paper is to clarify the relation between hoops and loops, or in other words, to give a description of the class of holonomy equivalent curves. | Groups of loops and hoops | 14,283 |
We define a C^1 distance between submanifolds of a riemannian manifold M and show that, if a compact submanifold N is not moved too much under the isometric action of a compact group G, there is a G-invariant submanifold C^1-close to N. The proof involves a procedure of averaging nearby submanifolds of riemannian manifolds in a symmetric way. The procedure combines averaging techniques of Cartan, Grove/Karcher, and de la Harpe/Karoubi with Whitney's idea of realizing submanifolds as zeros of sections of extended normal bundles. | Almost invariant submanifolds for compact group actions | 14,284 |
We consider the notion of stable isomorphism of bundle gerbes. It has the consequence that the stable isomorphism classes of bundle gerbes over a manifold M are in bijective correspondence with H^3(M, Z). Stable isomorphism sheds light on the local theory of bundle gerbes and enables us to develop a classifying theory for bundle gerbes using results of Gajer on BC^x bundles. | Bundle gerbes: stable isomorphism and local theory | 14,285 |
We show that a noncompact manifold with bounded sectional curvature, whose ends are sufficiently Gromov-Hausdorff close to rays, has a finite dimensional space of square-integrable harmonic forms. In the special case of a finite-volume manifold with pinched negative sectional curvature, we show that the essential spectrum of the p-form Laplacian is the union of the essential spectra of a collection of ordinary differential operators associated to the ends. We give examples of such manifolds with curvature pinched arbitrarily close to -1 and with an infinite number of gaps in the spectrum of the function Laplacian. | On the spectrum of a finite-volume negatively-curved manifold | 14,286 |
A result of Jost and Zuo is used to show that for a large class of finite-dimensional hyperk\"ahler quotients, the only L2 harmonic forms lie in the middle dimension, and are of type (k,k) with respect to all complex structures. The argument is extended to some moduli spaces which appear as infinite-dimensional quotients such as Higgs bundle and SU(2) monopole moduli spaces. In the latter case we recover some of the S-duality predictions of Sen. | L^2 cohomology of hyperkähler quotients | 14,287 |
We study Lagrangian subalgebras of a semisimple Lie algebra with respect to the imaginary part of the Killing form. We show that the variety $\Lagr$ of Lagrangian subalgebras carries a natural Poisson structure $\Pi$. We determine the irreducible components of $\Lagr$, and we show that each irreducible component is a smooth fiber bundle over a generalized flag variety, and that the fiber is the product of the real points of a De Concini-Procesi compactification and a compact homogeneous space. We study some properties of the Poisson structure $\Pi$ and show that it contains many interesting Poisson submanifolds. | On the variety of Lagrangian subalgebras | 14,288 |
It has recently been conjectured that the eigenvalues $\lambda$ of the Dirac operator on a closed Riemannian spin manifold $M$ of dimension $n\ge 3$ can be estimated from below by the total scalar curvature: $$ \lambda^2 \ge \frac{n}{4(n-1)} \cdot \frac{\int_M S}{vol(M)}. $$ We show by example that such an estimate is impossible. | Dirac eigenvalues and total scalar curvature | 14,289 |
A new technique for the study of geodesic connectedness in a class of Lorentzian manifolds is introduced. It is based on arguments of Brouwer's topological degree for the solution of functional equations. It is shown to be very useful for multiwarped spacetimes, which include different types of relativistic spacetimes. | Geodesic connectedness of multiwarped spacetimes | 14,290 |
We show that the contact reduction can be specialized to Sasakian manifolds. We link this Sasakian reduction to K\"ahler reduction by considering the K\"ahler cone over a Sasakian manifold. We present examples of Sasakian manifolds obtained by $S^1$ reduction of standard Sasakian spheres. | Reduction of Sasakian manifolds | 14,291 |
The paper concerns discrete versions of the three well-known results of projective differential geometry: the four vertex theorem, the six affine vertex theorem and the Ghys theorem on four zeroes of the Schwarzian derivative. We study geometry of closed polygonal lines in $\bbRP^d$ and prove that polygons satisfying a certain convexity condition have at least d+1 flattenings. This result provides a new approach to the above mentioned classical theorems. | Projective geometry of polygons and discrete 4-vertex and 6-vertex
theorems | 14,292 |
We find the shape of the Donaldson invariants of a 4-manifold with b_1=0 and b^+>1, which may be not of simple type. The invariants appear as the q^0 coefficient of a expression given in terms of modular forms (as was predicted by Moore and Witten). We re-express the formula using complete elliptic integrals to prove a conjecture of Kronheimer and Mrowka on the Donaldson invariants of non-simple type 4-manifolds. | Donaldson invariants of non-simple type 4-manifolds | 14,293 |
This paper achieves, among other things, the following: 1)It frees the main result of [BFKM] from the hypothesis of determinant class and extends this result from unitary to arbitrary representations. 2)It extends (and at the same times provides a new proof of) the main result of Bismut and Zhang [BZ] from finite dimensional representations of $\Gamma$ to representations on an ${\cal A}-$Hilbert module of finite type (${\cal A}$ a finite von Neumann algebra). The result of [BZ] corresponds to ${\cal A}=\bbc.$ 3)It provides interesting real valued functions on the space of representations of the fundamental group $\Gamma$ of a closed manifold M. These functions might be a useful source of topological and geometric invariants of M. These objectives are achieved with the help of the relative torsion $\cal R $, first introduced by Carey, Mathai and Mishchenko [CMM] in special cases. The main result of this paper calculates explicitly this relative torsion (cf Theorem 0.1). | Relative torsion | 14,294 |
For any closed complex manifold $X$, we calculate the Poincar\'{e} and Hodge polynomials of the delocalized equivariant cohomology $H^*(X^n, S_n)$ with a grading specified by physicists. As a consequence, we recover a special case of a formula for the elliptic genera of symmetric products in Dijkgraaf-Moore-Verlinde-Verlinde \cite{Dij-Moo-Ver-Ver}. For a projective surface X, our results matches with the corresponding formulas for the Hilbert scheme of X^[n]. We also give geometric construction of an action of a Heisenberg superalgebra on $\sum_{n \geq 0} H^{*,*}(X^n, S_n)$, imitating the constructions for equivariant K-theory by Segal \cite{Seg} and Wang \cite{Wan}. There is a corresponding version for $H^{-*, *}$. | Delocalized equivariant coholomogy of symmetric products | 14,295 |
We calculate the Hirzebruch $\chi_y$ and $\hat{\chi}_y$-genera of symmetric products of closed complex manifolds by the holomorphic Lefschetz formula of Atiyah and Singer \cite{Ati-Sin}. Such calculation rederive some formulas proved in an earlier paper \cite{Zho} by a different method. | Calculations of the Hirzebruch $χ_y$ genera of symmetric products by
the holomorphic Lefschetz formula | 14,296 |
An upper bound on the first S^1 invariant eigenvalue of the Laplacian for invariant metrics on the 2-sphere is used to find obstructions to the existence of isometric embeddings of such metrics in (R^3,can). As a corollary we prove: If the first four distinct eigenvalues have even multiplicities then the surface of revolution cannot be isometrically embedded in (R^3,can). This leads to a generalization of a classical result in the theory of surfaces. | The spectrum and isometric embeddings of surfaces of revolution | 14,297 |
In this paper, we first prove a local family version of the Atiyah-Bott-Segal-Singer Lefschetz fixed point formula, then we extend the famous Witten's rigidity Theorems to the family case. Several family vanishing theorems for elliptic genera are also proved. | On Family Rigidity Theorems I | 14,298 |
This article is a report on the status of the problem of classifying the irriducibly acting subgroups of GL(n,R) that can appear as the holonomy of a torsion-free affine connection. In particular, it contains an account of the completion of the classification of these groups by Chi, Merkulov, and Schwachhofer as well as of the exterior differential systems analysis that shows that all of these groups do, in fact, occur. Some discussion of the results of Joyce on the existence of compact examples with holonomy G_2 or Spin(7) is also included. | Recent Advances in the Theory of Holonomy | 14,299 |
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