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We give new and rather general gluing theorems for anti-self-dual (ASD) conformal structures, following the method suggested by Floer. The main result is a gluing theorem for pairs of conformally ASD manifolds `joined' across a common piece (union of connected components) of their boundaries. This theorem genuinely operates in the b-category (in the sense of Melrose) and in general the boundary of the joined manifold can be non-empty. The resulting metric is a conformally ASD b-metric or, in more traditional language, a complete conformally ASD metric with cylindrical asymptotics. We also study hermitian-ASD conformal structures on complex surfaces in relation to scalar-flat K\"ahler geometry. The general results are illustrated with a simple application, showing that the blow-up of C^2 at an arbitrary finite set of points admits scalar-flat K\"ahler metrics that are asymptotic to the Euclidean metric at infinity. A number of vanishing theorems for the obstruction space is also included. | Gluing theorems for complete anti-self-dual spaces | 14,400 |
We study Hamiltonian stationary Lagrangian surfaces in C^2, i.e. Lagrangian surfaces in C^2 which are stationary points of the area functional under smooth Hamiltonian variations. Using loop groups, we propose a formulation of the equation as a completely integrable system. We construct a Weierstrass type representation and produce all tori through either the integrable systems machinery or more direct arguments. | Hamiltonian stationary Lagrangian surfaces in C^2 | 14,401 |
The main result is the construction of ergodic transversal measures of full support on the space of all k-surfaces of a compact hyperbolic 3-manifold. This space is a laminated space, each of its leaf being identified with a "complete" k-surface, i.e. a surface of constant (extrinsic) curvature k, where k belongs to ]0,1[. Elsewhere, this space has been shown to have chaotic properties mimicking those of the geodesic flow. This construction is achieved through a coding of the space. | Random k-Surfaces | 14,402 |
Let X be a smooth, complete, connected submanifold of dimension n < N in a complex affine space A^N (C), and r is the rank of its Gauss map \gamma, \gamma (x) = T_x (X). The authors prove that if 2 \leq r \leq n - 1, N - n \geq 2, and in the pencil of the second fundamental forms of X, there are two forms defining a regular pencil all eigenvalues of which are distinct, then the submanifold X is a cylinder with (n-r)-dimensional plane generators erected over a smooth, complete, connected submanifold Y of rank r and dimension r. This result is an affine analogue of the Hartman-Nirenberg cylinder theorem proved for X \subset R^{n+1} and r = 1. For n \geq 4 and r = n - 1, there exist complete connected submanifolds X \subset A^N (C) that are not cylinders. | An affine analogue of the Hartman-Nirenberg cylinder theorem | 14,403 |
In this paper, we study the weak compactness of the set of conformal metrics in any Riemann surface without boundary whose Calabi energy and area are uniformly bounded. We prove that for any sequence of such metrics, there alwasy exists a subsequence which converges in H\sp{2,2}_\sb{loc} everywhere except a finite number of bubble points. Blowup analysis near bubble shows that the bubble on bubble phenomenon occurs. The limit metric gives rise to a tree structure decomposition, where each node in the tree represents a limit metric of a subsequence at that stage while the edge of the tree structure represents the neck on the process of blowing up. We also show that the number of the nodes which have more than three edges attached is finite. | Weak Limits of Riemannian Metrics in Surfaces with integral Curvature
Bound | 14,404 |
In this paper, we observe a set of functionals of metrics which are all decrease under the Calabi flow and have uniform lower bound along the flow, which give rise to a set of integral estimates on the curvature flow. Using these estimates, together with weak compactness we obtained in previous papers [8] and [10], we prove the long term existence and convergence of the Calabi flow. Thus give a new proof to Chruscial's theorem. The set of simple ideas of global integral estimates and concentration compactness should have further implications in other heat flow problems. | Calabi flow in Riemann surfaces revisited: A new point of view | 14,405 |
In this paper, we introduce a new parabolic equation on K\"ahler manifolds. The static point of this flow is related to the existence of a lower bound of the Mabuchi energy. In this paper, we prove the flow always exists for all times for any initial smooth data. Further more, if the initial metric has non-negative bisectional curvature, we prove the flow converges to a static metric eventually. | A new parabolic flow in Kaehler manifolds | 14,406 |
In this paper, we announce the following results: Let M be a Kaehler-Einstein manifold with positive scalar curvature. If the initial metric has nonnegative bisectional curvature and positive at least at one point, then the K\"ahler-Ricci flow converges exponentially fast to a Kaehler-Einstein metric with constant bisectional curvature. | Ricci flow on Kaehler manifolds | 14,407 |
In this paper, we construct a set of new functionals of Ricci curvature on any Kaehler manifolds which are invariant under holomorphic transfermations in Kaehler Einstein manifolds and essentially decreasing under the Kaehler Ricci flow. Moreover, if the initial metric has non-negative bisectional curvature, using Tian's inequality, we can prove that each of the functionals has uniform lower bound along the flow which gives a set of integral estimates on curvature. Using this set of integral estimates, we are able to show the following theorem: Let M be a Kaehler-Einstein surface with positive scalar curvature. If the initial metric has nonnegative sectional curvature and positive somewhere, then the Kaehler-Ricci flow converges exponentially fast to a Kaehler-Einstein metric with constant bisectional curvature. | Ricci flow on Kaehler-Einstein surfaces | 14,408 |
The geometry of submanifolds is intimately related to the theory of functions and vector bundles. It has been of fundamental importance to find out how those two objects interact in many geometric and physical problems. A typical example of this relation is that the Picard group of line bundles on an algebraic manifold is isomorphic to the group of divisors, which is generated by holomorphic hypersurfaces modulo linear equivalence. A similar correspondence can be made between the K-group of sheaves and the Chow ring of holomorphic cycles. There are two more very recent examples of such a relation. The mirror symmetry in string theory has revealed a deeper phenomenon involving special Lagrangian cycles (cf. [SYZ]). On the other hand, C. Taubes has shown that the Seiberg-Witten invariant coincides with the Gromov-Witten invariant on any symplectic 4-manifolds. In this paper, we will show another natural interaction between Yang-Mills connections, which are critical points of a Yang-Mills action associated to a vector bundle, and minimal submanifolds, which have been studied extensively for years in classical differential geometry and the calculus of variations. | Gauge theory and calibrated geometry, I | 14,409 |
This is the third in a series of papers constructing explicit examples of special Lagrangian submanifolds in C^m. The previous paper in the series, math.DG/0008155, defined the idea of evolution data, which includes an (m-1)-submanifold P in R^n, and constructed a family of special Lagrangian m-folds N in C^m, which are swept out by the image of P under a 1-parameter family of linear or affine maps phi_t : R^n -> C^m, satisfying a first-order o.d.e. in t. In this paper we use the same idea to construct special Lagrangian 3-folds in C^3. We find a 1-1 correspondence between sets of evolution data with m=3 and homogeneous symplectic 2-manifolds P. This enables us to write down several interesting sets of evolution data, and so to construct corresponding families of special Lagrangian 3-folds in C^3. Our main results are a number of new families of special Lagrangian 3-folds in C^3, which we write very explicitly in parametric form. Generically these are nonsingular as immersed 3-submanifolds, and diffeomorphic to R^3 or S^1 x R^2. Some of the 3-folds are singular, and we describe their singularities, which we believe are of a new kind. We hope these 3-folds will be helpful in understanding singularities of compact special Lagrangian 3-folds in Calabi-Yau 3-folds. This will be important in resolving the SYZ conjecture in Mirror Symmetry. | Evolution equations for special Lagrangian 3-folds in C^3 | 14,410 |
We consider the sigma models where the base metric is proportional to the metric of the configuration space. We show that the corresponding sigma model equation admits a Lax pair. We also show that this type of sigma models in two dimensions are intimately related to the minimal surfaces in a flat pseudo Riemannian 3-space. We define two dimensional surfaces conformally related to the minimal surfaces in flat three dimensional geometries which enable us to give a construction of the metrics of some even dimensional Ricci flat pseudo Riemannian geometries. | Sigma Models, Minimal Surfaces and Some Ricci Flat Pseudo Riemannian
Geometries | 14,411 |
We find three characterizations for a multidimensional (n+1)-web W possessing a reduct reducible subweb: its closed form equations, the integrability of an invariant distribution associated with W, and the relations between the components of its torsion tensor. In the case of codimension one, the latter criterion establishes a relation with solutions of a system of nonlinear second-order PDEs. Some particular cases of this system were considered by Goursat in 1899. | Multidimensional (n+1)-Webs with Reduct Reducible Subwebs | 14,412 |
We consider the Goursat's (n+1)-webs of codimension one of two kinds on an n-dimensional manifold. They are characterized by the specific closed form equations or by two special relations between components of the torsion tensor of the web. These relations allow us to establish a connection with solutions of two systems of nonlinear second-order PDEs investigated by Goursat in 1899. The integrability conditions of some distributions invariantly associated with both kinds of Goursat's (n+1)-webs are also investigated. | Goursat's (n+1)-Webs | 14,413 |
The authors study smooth lines on projective planes over the algebra C of complex numbers, the algebra C^1 of double numbers, and the algebra C^0 of dual numbers. In the space RP^5, to these smooth lines there correspond families of straight lines describing point three-dimensional tangentially degenerate submanifolds X^3 of rank 2. The authors study focal properties of these submanifolds and prove that they represent examples of different types of tangentially degenerate submanifolds. Namely, the submanifold X^3, corresponding in RP^5 to a smooth line \gamma of the projective plane C, does not have real singular points, the submanifold X^3, corresponding in RP^5 to a smooth line \gamma of the projective plane C^1 P^2, bears two plane singular lines, and finally the submanifold X^3, corresponding in RP^5 to a smooth line \gamma of the projective plane C^0 P^2, bears one singular line. | Geometry of Projective Planes over Two-Dimensional Algebras | 14,414 |
We establish extremality of Riemannian metrics g with non-negative curvature operator on symmetric spaces M=G/K of compact type with rk(G)-rk(K)\le 1. Let g' be another metric with scalar curvature k', such that g'\ge g on 2-vectors. We show that k'\ge k everywhere on M implies k'=k. Under an additional condition on the Ricci curvature of g, k'\ge k even implies g'=g. We also study area-non-increasing spin maps onto such Riemannian manifolds. | Scalar curvature estimates for compact symmetric spaces | 14,415 |
The purpose of this note is to establish the following theorem: Let N be a Kahler manifold, L be a compact oriented immersed minimal Lagrangian submanifold in N and V be a holomorphic vector field in a neighbourhood of L in N. Let div(V) be the (complex) divergence of V. Then the integral of div(V) over L is 0. Vice versa let N^2n be Kahler-Einstein with non-zero scalar curvature and L^n be a totally real oriented embedded n-dimensional real-analytic submanifold of N s.t. the divergence of any holomorphic vector field defined in a neighbourhood of L in N integrates to 0 on L. Then L is a minimal Lagrangian submanifold of N. | Holomorphic vector fields and minimal Lagrangian submanifolds | 14,416 |
We obtain new families of (1,2)-symplectic invariant metrics on the full complex flag manifolds F(n). For n > 4, we characterize n-3 different n-dimensional families of (1,2)-symplectic invariant metrics on F(n). Any of these families corresponds to a different class of non-integrable invariant almost complex structure on F(n). | Families of (1,2)-Symplectic M etrics on Full Flag Manifolds | 14,417 |
This paper is the third of a series on Hamiltonian stationary Lagrangian surfaces. We present here the most general theory, valid for any Hermitian symmetric target space. Using well-chosen moving frame formalism, we show that the equations are equivalent to an integrable system, generalizing the C^2 subcase analyzed in the first article (arXiv:math.DG/0009202). This system shares many features with the harmonic map equation of surfaces into symmetric spaces, allowing us to develop a theory close to Dorfmeister, Pedit and Wu's, including for instance a Weierstrass-type representation. Notice that this article encompasses the article mentioned above, although much fewer details will be given on that particular flat case. | Hamiltonian stationary Lagrangian surfaces in Hermitian symmetric spaces | 14,418 |
In an earlier paper, we established a natural connection between the Baum-Connes conjecture and noncommutative Bloch theory, viz. the spectral theory of projectively periodic elliptic operators on covering spaces. We elaborate on this connection here and provide significant evidence for a fundamental conjecture in noncommutative Bloch theory on the non-existence of Cantor set type spectrum. This is accomplished by establishing an explicit lower bound for the Kadison constant of twisted group C*-algebras in a large number of cases, whenever the multiplier is rational. | On positivity of the Kadison constant and noncommutative Bloch theory | 14,419 |
The paper developes a geometrization of a Kronecker $h$-regular vertical fundamental metrical d-tensor $G^{(\alpha)(\beta)}_{(i)(j)}$ on the jet fibre bundle of order one $J^1(T,M)$. This geometrization gives a mathematical model for both gravitational and electromagnetic field theory, in a general setting. In this context, the Einstein and Maxwell equations are described. | Generalized Metrical Multi-Time Lagrange Geometry of Physical Field | 14,420 |
Helfer in [Pacific J. Math. 164/2 (1994), p. 321--350] was the first to produce an example of a spacelike Lorentzian geodesic with a continuum of conjugate points. In this paper we show the following result: given an interval $[a,b]$ of $IR$ and any closed subset $F$ of $IR$ contained in $]a,b]$, then there exists a Lorentzian manifold $(M,g)$ and a spacelike geodesic $\gamma:[a,b]\to M$ such that $\gamma(t)$ is conjugate to $\gamma(a)$ along $\gamma$ iff $t\in F$. | On the Distribution of Conjugate Points along semi-Riemannian Geodesics | 14,421 |
This paper presents a natural extension to foliated spaces of the following result due to Gromov : the h-principle for open, invariant differential relations is valid on open manifolds. The definition of openness for foliated spaces adopted here involves a certain type of Morse functions. Consequences concerning the problem of existence of regular Poisson structures, the original motivation for this work, are presented. | A h-principle for open relations invariant under foliated isotopies | 14,422 |
We characterize compact locally conformally K\"ahler (l.c.K.) manifolds under the assumption of a purely conformal, holomorphic circle action. As an application, we determine the structure of the compact l.c.K. manifolds with parallel Lee form. We introduce the Lee-Cauchy-Riemann (LCR) transformations as a class of diffeomorphisms preserving the specific $G$-structure of l.c.K. manifolds. Then we characterize the Hopf manifolds, up to holomorphic isometry, as compact l.c.K. manifolds admitting a certain closed LCR action of $\mathbb{C}^*$. | Transformations of compact locally conformally Kähler manifolds | 14,423 |
This paper relates the boundary term in the Chern-Gauss-Bonnet formula on 4-manifolds M with the renormalized volume V, as defined in the AdS/CFT correspondence, for asymptotically hyperbolic Einstein metrics on M. In addition, we compute and discuss the differential or variation dV of V, or equivalently the variation of the L^2 norm of the Weyl curvature, on the space of such Einstein metrics. | L^2 curvature and volume renormalization of AHE metrics on 4-manifolds | 14,424 |
We give a simple proof of the local version of a result of R. Bryant, stating that any 3-dimensional Riemannian manifold can be isometrically embedded as a special Lagrangian submanifold in a Calabi-Yau manifold. We refine the theorem proving that a certain class of one-parameter families of metrics on a 3-torus can be isometrically embedded in a Calabi-Yau manifold as a one-parameter family of special Lagrangian submanifolds. We use our examples of one-parameter families to show that the semi-flat metric on the mirror manifold proposed be N. Hitchin is not necessarily Ricci-flat in dimension 3. | Some families of special Lagrangian tori | 14,425 |
We prove a semi-Riemannian version of the celebrated Morse Index Theorem for geodesics in semi-Riemannian manifolds; we consider the general case of both endpoints variable on two submanifolds. The key role of the theory is played by the notion of the {\em Maslov index} of a semi-Riemannian geodesic, which is a homological invariant and it substitutes the notion of geometric index in Riemannian geometry. Under generic circumstances, the Maslov index of a geodesic is computed as a sort of {\em algebraic count} of the conjugate points along the geodesic. For non positive definite metrics the index of the index form is always infinite; in this paper we prove that the space of all variations of a given geodesic has a {\em natural} splitting into two infinite dimensional subspaces, and the Maslov index is given by the difference of the index and the coindex of the restriction of the index form to these subspaces. In the case of variable endpoints, two suitable correction terms, defined in terms of the endmanifolds, are added to the equality. Using appropriate change of variables, the theory is entirely extended to the more general case of {\em symplectic differential systems}, that can be obtained as linearizations of the Hamilton equations. | The Morse Index Theorem in semi-Riemannian Geometry | 14,426 |
We introduce an approach based on moving frames for polygon recognition and symmetry detection. We present detailed algorithms for recognition of polygons modulo the special Euclidean, Euclidean, equi-affine, skewed-affine and similarity Lie groups, and explain the procedure for a generic Lie group. The time complexity of our algorithms is linear in the number of vertices and they are noise resistant. The signatures used allow the detection of partial as well as approximate equivalences. Our method is a particular case of a general method for curve recognition modulo Lie group action. | Polygon Recognition and Symmetry Detection | 14,427 |
The paper construct a suitable generalized metrical multi-time Lagrange geometrical model for both gravitational and electromagnetic fields, in a general setting. In this construction, the gravitational potentials are described by a distinguished vertical metrical tensor of the form $h^{/alpha/beta}e^{2\sigma}/phi_{ij}$. | Generalized Metrical Multi-Time Lagrange Model for General Relativity
and Electromagnetism | 14,428 |
In this note we prove that whenever a Lie group $G$ acts on a manifold $X$, then the orbit $Gx$ through any point $x$ of $X$ is a weakly embedded submanifold of $X$. The investigation of this problem was inspired by an application to Cat astrophe Theory. | Orbits of Lie Group Actions are Weakly Embedded | 14,429 |
The Newman-Penrose-Perjes formalism is applied to smooth contact structures on riemannian 3-manifolds. In particular it is shown that a contact 3-manifold admits an adapted riemannian metric if and only if it admits a metric with a divergence-free, constantly twisting, geodesic congruence. The shear of this congruence is identified with the torsion of the associated pseudohermitian structure, while the Tanaka-Webster curvature is identified with certain derivatives of the spin coefficients. The particular case where the associated riemannian metric is Einstein is studied in detail. It is found that the torsion is constant and the field equations are completely solved locally. Hyperbolic space forms are shown not to have adapted contact structures, even locally, while contact structures adapted to a flat or elliptic space form are contact isometric to the standard one. | Einstein Metrics Adapted to Contact Structures on 3-Manifolds | 14,430 |
This is the fourth in a series of papers math.DG/0008021, math.DG/0008155, math.DG/0010036 constructing explicit examples of special Lagrangian submanifolds (SL m-folds) in C^m. A submanifold of C^m is ruled if it is fibred by a family of real straight lines in C^m. This paper studies ruled special Lagrangian 3-folds in C^3, giving both general theory and families of examples. Our results are related to previous work of Harvey and Lawson, Borisenko and Bryant. An important class of ruled SL 3-folds is the special Lagrangian cones in C^3. Each ruled SL 3-fold is asymptotic to a unique SL cone. We study the family of ruled SL 3-folds N asymptotic to a fixed SL cone N_0. We find that this depends on solving a linear equation, so that the family of such N has the structure of a vector space. We also show that the intersection Sigma of N_0 with the unit sphere in C^3 is a Riemann surface, and construct a ruled SL 3-fold N asymptotic to N_0 for each holomorphic vector field w on Sigma. As corollaries of this we write down two large families of explicit SL 3-folds depending on a holomorphic function on C, which include many new examples of singularities of SL 3-folds. We also show that each SL T^2 cone N_0 can be extended to a 2-parameter family of ruled SL 3-folds asymptotic to N_0, and diffeomorphic to T^2 x R. | Ruled special Lagrangian 3-folds in C^3 | 14,431 |
It has long been known that differential forms on complex manifolds can be decomposed under the action of the complex structure to give the Dolbeault complex. This paper presents an analogous double complex for quaternionic manifolds using the fact that the cotangent space is isomorphic to a quaternionic vector space. This defines an action of the group Sp(1) of unit quaternions on the cotangent space, which induces an action of Sp(1) on the space of k-forms. A double complex is obtained by decomposing the k-forms into irreducible representations of Sp(1), resulting in new 'quaternionic Dolbeault' operators and cohomology groups. Links with previous work in quaternionic geometry, particularly the differential complex of Salamon and the q-holomorphic functions of Joyce, are demonstrated. | A Dolbeault-type Double Complex on Quaternionic Manifolds | 14,432 |
We give a new, connected-sum-like construction of Riemannian metrics with special holonomy G_2 on compact 7-manifolds. The construction is based on a gluing theorem for appropriate elliptic partial differential equations. As a prerequisite, we also obtain asymptotically cylindrical Riemannian manifolds with holonomy SU(3) building up on the work of Tian and Yau. Examples of new topological types of compact 7-manifolds with holonomy G_2 are constructed using Fano 3-folds. | Twisted connected sums and special Riemannian holonomy | 14,433 |
We consider systems $(M,\omega,g)$ with $M$ a closed smooth manifold, $\omega$ a real valued closed one form and $g$ a Riemannian metric, so that $(\omega,g)$ is a Morse-Smale pair, Definition~2. We introduce a numerical invariant $\rho(\omega,g)\in[0,\infty]$ and improve Morse-Novikov theory by showing that the Novikov complex comes from a cochain complex of free modules over a subring $\Lambda'_{[\omega],\rho}$ of the Novikov ring $\Lambda_{[\omega]}$ which admits surjective ring homomorphisms $\ev_s:\Lambda'_{[\omega],\rho}\to\C$ for any complex number $s$ whose real part is larger than $\rho$. We extend Witten-Helffer-Sj\"ostrand results from a pair $(h,g)$ where $h$ is a Morse function to a pair $(\omega,g)$ where $\omega$ is a Morse one form. As a consequence we show that if $\rho<\infty$ the Novikov complex can be entirely recovered from the spectral geometry of $(M,\omega,g)$. | On the topology and analysis of a closed one form. I (Novikov's theory
revisited) | 14,434 |
Let $M$ be either a projective manifold $(M,Pi)$ or a pseudo-Riemannian manifold $(M,g).$ We extend, intrinsically, the projective/conformal Schwarzian derivatives that we have introduced recently, to the space of differential operators acting on symmetric contravariant tensor fields of any degree on $M.$ As operators, we show that the projective/conformal Schwarzian derivatives depend only on the projective connection $Pi$ and the conformal class $[g]$ of the metric, respectively. Furthermore, we compute the first cohomology group of $Vect(M)$ with coefficients into the space of symmetric contravariant tensor fields valued into $delta$-densities as well as the corresponding relative cohomology group with respect to $sl(n+1,R).$ | Projective and Conformal Schwarzian Derivatives and Cohomology of Lie
Algebras Vector Fields Related to Differential Operators | 14,435 |
We construct metrics of positive Ricci curvature on some vector bundles over tori (or more generally, over nilmanifolds). This gives rise to the first examples of manifolds with positive Ricci curvature which are homotopy equivalent but not homeomorphic to manifolds of nonnegative sectional curvature. | Metrics of positive Ricci curvature on vector bundles over nilmanifolds | 14,436 |
We give new estimates for the eigenvalues of the hypersurface Dirac operator in terms of the intrinsic energy-momentum tensor, the mean curvature and the scalar curvature. We also discuss their limiting cases as well as the limiting cases of the estimates obtained by X. Zhang and O. Hijazi in [13] and [10]. We compare these limiting cases with those corresponding to the Friedrich and Hijazi inequalities. We conclude by comparing these results to intrinsic estimates for the Dirac-Schr\"odinger operator D_f = D - f/2. | Eigenvalue estimates for the Dirac-Schrödinger operators | 14,437 |
We construct several natural connections and Dirac type operators on a general metric contact manifold which are more sensitive to the geometric background. In the special case of CR manifolds these connections are also compatible with the CR structure and include among them the Webster connection. We also describe several Weitzenbock type formulae. Our method is based on work of P. Gauduchon applied to two almost complex manifolds naturally associated to a given metric contact manifold. | Geometric connections and geometric Dirac operators on contact manifolds | 14,438 |
We deal with compact Kaehler manifolds M which are acted on by a semisimple compact Lie group G of isometries with codimension one regular orbits. We provide an explicit description of the standard blow-ups of such manifolds along complex singular orbits, in case b_1(M) = 0 and the regular orbits are Levi nondegenerate. Up to very few exceptions, all the nonhomogeneous manifolds in this class are shown to admit a G-invariant Kaehler-Einstein metric, giving completely new examples of compact Kaehler-Einstein manifolds. | Running after a new Kaehler-Einstein metric | 14,439 |
Bott and Samuelson constructed explicit cycles representing a basis of the Z_2-homology of the orbits of variationally complete representations of compact Lie groups. As a consequence, all those orbits are taut. We were able to show that an irreducible representation of a compact Lie group, all of whose orbits are taut, is either variationally complete or it is one of the following orthogonal representations (n bigger than or equal to 2): the (standard) x_R (spin) representation of SO(2)xSpin(9); or the (standard) x_C (standard) representation of U(2)xSp(n); or the (standard)^3 x_ H (standard) representation of SU(2)xSp(n). In this paper we will show how to adapt the construction of the cycles of Bott and Samelson to the orbits of these three representations. As a result, they also admit explicit cycles representing a basis of their Z_2-homology and, in particular, this provides another proof of their tautness. | Cycles of Bott-Samelson type for taut representations | 14,440 |
In this paper we prove that every H-type Lie algebra possesses a basis with respect to which the structure constants are integers. Existence of such an integral basis implies via the Mal'cev criterion that all simply connected H-type Lie groups contain cocompact lattices. Since the Campbell-Hausdorff formula is very simple for two-step nilpotent Lie groups we can actually avoid invoking the Mal'cev criterion and exhibit our lattices in an explicit way. As an application, we calculate the isoperimetric dimensions of H-type groups. | Integral Structures on H-type Lie Algebras | 14,441 |
This text proposes geometrical descriptions of all variational problems invariant by conformal transformations in two variables. First a characterisation in terms of C-Finsler manifolds, a suitable generalization of Finsler manifolds, is given. Second Hamiltonian formalisms are explored, with an emphasis on Caratheodory's formalism. Some of these considerations were subsequently developped in papers with Joseph Kouneiher in math-ph/0004020 and math-ph/0010036 | Problemes variationnels invariants par transformation conforme en
dimension 2 | 14,442 |
This is the sixth in a series of papers constructing examples of special Lagrangian m-folds in C^m. We present a construction of special Lagrangian cones in C^3 involving two commuting o.d.e.s, motivated by the first two papers of the series. Then we generalize it to a construction of non-conical special Lagrangian 3-folds in C^3 involving three commuting o.d.e.s. Now special Lagrangian cones in C^3 are linked to the theory of harmonic maps and integrable systems. Harmonic maps from a Riemann surface into complex projective space CP^n are an integrable system, and can be studied and classified using loop group techniques. If N is a special Lagrangian cone in C^3, then N is the cone on the image of a conformal harmonic map \psi : S --> S^5 for some Riemann surface S, and the projection of \psi to CP^2 is also conformal harmonic. Our examples of special Lagrangian cones in C^3 yield conformal harmonic maps \psi : R^2 --> CP^2. We work through the integrable systems theory for these examples, showing that they are superconformal of finite type, and calculating their harmonic sequences, Toda and Tzitzeica solutions, algebra of polynomial Killing fields and spectral curves. We also study the double periodicity conditions for \psi, and so find families of superconformal tori in CP^2. We finish by asking whether our more general construction of special Lagrangian 3-folds can also be derived from a higher-dimensional integrable system, and whether the special Lagrangian equations themselves are in some sense integrable. | Special Lagrangian 3-folds and integrable systems | 14,443 |
In this paper we present a new approach to Morse theory based on the de Rham-Federer theory of currents. The full classical theory is derived in a transparent way. The methods carry over uniformly to the equivariant and the holomorphic settings. Moreover, the methods are substantially stronger than the classical ones and have interesting applications to geometry. They lead, for example, to formulae relating characteristic forms and singularities of bundle maps. | Finite volume flows and Morse theory | 14,444 |
The Newman-Penrose-Perjes formalism is applied to Sasakian 3-manifolds and the local form of the metric and contact structure is presented. The local moduli space can be parameterised by a single function of two variables and it is shown that, given any smooth function of two variables, there exists locally a Sasakian structure with scalar curvature equal to this function. The case where the scalar curvature is constant ($\eta$-Einstein Sasakian metrics) is completely solved locally. The resulting Sasakian manifolds include $S^3$, $Nil$ and $\tilde{SL_2R}$, as well as the Berger spheres. It is also shown that a conformally flat Sasakian 3-manifold is Einstein of positive scalar curvature. | The Local Moduli of Sasakian 3-Manifolds | 14,445 |
In this work, complete constant mean curvature 1 (CMC-1) surfaces in hyperbolic 3-space with total absolute curvature at most 4 pi are classified. This classification suggests that the Cohn-Vossen inequality can be sharpened for surfaces with odd numbers of ends, and a proof of this is given. | Mean curvature 1 surfaces in hyperbolic 3-space with low total curvature
II | 14,446 |
In the previous paper, Takahasi and the authors generalized the theory of minimal surfaces in Euclidean n-space to that of surfaces with holomorphic Gauss map in certain class of non-compact symmetric spaces. It also includes the theory of constant mean curvature one surfaces in hyperbolic 3-space. Moreover, a Chern-Osserman type inequality for such surfaces was shown. Though its equality condition is not solved yet, the authors have noticed that the equality condition of the original Chern-Osserman inequality itself is not found in any literature except for the case n=3, in spite of its importance. In this paper, a simple geometric condition for minimal surfaces that attains equality in the Chern-Osserman inequality is given. The authors hope it will be a useful reference for readers. | Minimal surfaces that attain equality in the Chern-Osserman inequality | 14,447 |
We introduce a new quasi-isometry invariant $\subcorank X$ of a metric space $X$ called {\it subexponential corank}. A metric space $X$ has subexponential corank $k$ if roughly speaking there exists a continuous map $g:X\to T$ such that for each $t\in T$ the set $g^{-1}(t)$ has subexponential growth rate in $X$ and the topological dimension $\dim T=k$ is minimal among all such maps. Our main result is the inequality $\hyprank X\le\subcorank X$ for a large class of metric spaces $X$ including all locally compact Hadamard spaces, where $\hyprank X$ is maximal topological dimension of $\di Y$ among all $\CAT(-1)$ spaces $Y$ quasi-isometrically embedded into $X$ (the notion introduced by M. Gromov in a slightly stronger form). This proves several properties of $\hyprank$ conjectured by M. Gromov, in particular, that any Riemannian symmetric space $X$ of noncompact type possesses no quasi-isometric embedding $\hyp^n\to X$ of the standard hyperbolic space $\hyp^n$ with $n-1>\dim X-\rank X$. | Hyperbolic rank and subexponential corank of metric spaces | 14,448 |
We show that $\scriptstyle{#9(S^2\times S^3)}$ admits an 8-dimensional complex family of inequivalent non-regular Sasakian-Einstein structures. These are the first known Einstein metrics on this 5-manifold. In particular, the bound $\scriptstyle{b_2(M)\leq8}$ which holds for any regular Sasakian-Einstein $\scriptstyle{M}$ does not apply to the non-regular case. We also discuss the failure of the Hitchin-Thorpe inequality in the case of 4-orbifolds and describe the orbifold version. | Sasakian-Einstein Structures on $9#(S^2\times S^3)$ | 14,449 |
In 1841, Delaunay constructed the embedded surfaces of revolution with constant mean curvature (CMC); these unduloids have genus zero and are now known to be the only embedded CMC surfaces with two ends and finite genus. Here, we construct the complete family of embedded CMC surfaces with three ends and genus zero; they are classified using their asymptotic necksizes. We work in a class slightly more general than embedded surfaces, namely immersed surfaces which bound an immersed three-manifold, as introduced by Alexandrov. | Triunduloids: Embedded constant mean curvature surfaces with three ends
and genus zero | 14,450 |
We survey our recent results on classifying complete constant mean curvature 1 (CMC-1) surfaces in hyperbolic 3-space with low total curvature. There are two natural notions of "total curvature"-- one is the total absolute curvature which is the integral over the surface of the absolute value of the Gaussian curvature, and the other is the dual total absolute curvature which is the total absolute curvature of the dual CMC-1 surface. Here we discuss results on both notions (proven in two other papers by the authors), and we introduce some new results (with proofs) as well. | Period problems for mean curvature one surfaces in H^3 (with application
to surfaces of low total curvature) | 14,451 |
Geodesics and curvature of semidirect product groups with right invariant metrics are determined. In the special case of an isometric semidirect product, the curvature is shown to be the sum of the curvature of the two groups. A series of examples, like the magnetic extension of a group, are then considered. | Geodesics and curvature of semidirect product groups | 14,452 |
We classify the normal CR structures on $S^3$ and their automorphism groups. Together with [3], this closes the classification of normal CR structures on contact 3-manifolds. We give a criterion to compare 2 normal CR structures, and we show that the underlying contact structure is, up to homotopy, unique. | Automorphism groups of normal CR 3-manifolds | 14,453 |
We describe some general constructions on a real smooth projective 4-quadric which provide analogues of the Willmore functional and conformal Gauss map in both Lie sphere and projective differential geometry. Extrema of these functionals are characterized by harmonicity of this Gauss map. | Harmonic maps in unfashionable geometries | 14,454 |
We determine the Riemannian manifolds for which the group of exact volume preserving diffeomorphisms is a totally geodesic subgroup of the group of volume preserving diffeomorphisms, considering right invariant $L^2$-metrics. The same is done for the subgroup of Hamiltonian diffeomorphisms as a subgroup of the group of symplectic diffeomorphisms in the K\"ahler case. These are special cases of totally geodesic subgroups of diffeomorphisms with Lie algebras big enough to detect the vanishing of a symmetric 2-tensor field. | Totally geodesic subgroups of diffeomorphisms | 14,455 |
We establish an index theorem for Toeplitz operators on odd dimensional spin manifolds with boundary. It may be thought of as an odd dimensional analogue of the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary. In particular, there occurs naturally an invariant of $\eta$ type associated to $K^1$ representatives on even dimensional manifolds, which should be of independent interests. For example, it gives an intrinsic interpretation of the so called Wess-Zumino term in the WZW theory in physics. | An Index Theorem for Toeplitz Operators on Odd Dimensional Manifolds
with Boundary | 14,456 |
Let M be a closed compact n-dimensional manifold with n odd. We calculate the first and second variations of the zeta-regularized determinants det^\prime\Lambda and det L as the metric on M varies, where \Delta denotes the Laplacian on functions and L denotes the conformal Laplacian. We see that the behavior of these functionals denotes the conformal Laplacian. We see that the behavior of these functionals depends on the dimension. Indeed, every critical metric for (-1)^{(n-1)/2}det^\prime\Lambda or (-1)^{(n-1}/2}| det L| has finite index. Consequently there are no local maxima if n=4m+1 and no local minima if n=4m+3. We show that the standard 3-sphere is a local maximum for det^\prime\Lambda while the standard (4m-3)-sphere with m=1,2,...,4, is a saddle point. By contrast, for all odd n, the standard n-sphere is a local extremal for det L. An important tool in our work is the canonical trace on odd class operators in odd dimensions. This trace is related to the determinant by the formula det Q = TR log Q, and we prove some basic results on how to calculate the trace. | Critical metrics for the determinant of the Laplacian in odd dimensions | 14,457 |
We propose a twistor construction of surfaces in Lie sphere geometry based on the linear system which copies equations of Wilczynski's projective frame. In the particular case of Lie-applicable surfaces this linear system describes joint eigenfunctions of a pair of commuting Schr\"odinger operators with magnetic fields. | The analogue of Wilczynski's projective frame in Lie sphere geometry:
Lie-applicable surfaces and commuting Schrödinger operators with magnetic
fields | 14,458 |
A short proof of the Caratheodory conjecture about index of an isolated umbilic on the convex 2-dimensional sphere is suggested. The argument is based on the study of geodesic lines near cone-type singularity of a metric induced by holomorphic quadratic differentials. | CMC-surfaces, flat structures and umbilical points | 14,459 |
It was pointed out that the space of hermitian triples is an anology of the hermitian connection space. Generalizing the Ashtekar - Isham procedure one can quantize the space of hermitian triples as well as the original one. Here we add an example how this similarity can be exploited in a quantum theory of riemannian geometry. | The space of hermitian triples and the Ashtekar - Isham quantization | 14,460 |
We prove a new lower bound for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold by refined Weitzenb\"ock techniques. It applies to manifolds with harmonic curvature tensor and depends on the Ricci tensor. Examples show how it behaves compared to other known bounds. | Eigenvalue estimates of the Dirac operator depending on the Ricci tensor | 14,461 |
We show that Jacobi fields along harmonic maps between suitable spaces preserve conformality, holomorphicity, real isotropy and complex isotropy to first order; this last being one of the key tools in the proof by Lemaire and the author of integrability of Jacobi fields along harmonic maps from the 2-sphere to the complex projective plane. | Jacobi fields along harmonic maps | 14,462 |
A Sasakian structure on a manifold is called {\it positive} if its basic first Chern class can be represented by a positive (1,1)-form with respect to its transverse holomorphic CR-structure. We prove a theorem that says that every positive Sasakian structure can be deformed to a Sasakian structure whose metric has positive Ricci curvature. This allows us by example to give a completely independent proof of a result of Sha and Yang [SY] that for every positive integer k the k-fold connected sum of $S^2\times S^3$ admits metrics of positive Ricci curvature. | On Positive Sasakian Geometry | 14,463 |
This paper studies several aspects of asymptotically hyperbolic Einstein metrics, mostly on 4-manifolds. We prove boundary regularity (at infinity) for such metrics and establish uniqueness under natural conditions on the boundary data. By examination of explicit black hole metrics, it is shown that neither uniqueness nor finiteness holds in general for AH Einstein metrics with a prescribed conformal infinity. We then describe natural conditions which are sufficient to ensure finiteness. | Boundary regularity, uniqueness and non-uniqueness for AH Einstein
metrics on 4-manifolds | 14,464 |
We study a natural map from representations of a free group of rank g in GL(n,C), to holomorphic vector bundles of degree 0 over a compact Riemann surface X of genus g, associated with a Schottky uniformization of X. Maximally unstable flat bundles are shown to arise in this way. We give a necessary and sufficient condition for this map to be a submersion, when restricted to representations producing stable bundles. Using a generalized version of Riemann's bilinear relations, this condition is shown to be true on the subspace of unitary Schottky representations. | Schottky uniformization and vector bundles over Riemann surfaces | 14,465 |
We show that any Jacobi field along a harmonic map from the 2-sphere to the complex projective plane is integrable (i.e., is tangent to a smooth variation through harmonic maps). This provides one of the few known answers to this problem of integrability, which was raised in different contexts of geometry and analysis. It implies that the Jacobi fields form the tangent bundle to each component of the manifold of harmonic maps from $S^2$ to ${\bf C}P^2$ thus giving the nullity of any such harmonic map; it also has bearing on the behaviour of weakly harmonic $E$-minimizing maps from a 3-manifold to ${\bf C}P^2$ near a singularity and the structure of the singular set of such maps from any manifold to ${\bf C}P^2$. | Jacobi fields along harmonic 2-spheres in ${\bf C}P^2$ are integrable | 14,466 |
Let ${\cal F}_\lambda$ be the space of tensor densities on ${\bf R}^n$ of degree $\lambda$ (or, equivalently, of conformal densities of degree $-\lambda{}n$) considered as a module over the Lie algebra $so(p+1,q+1)$. We classify $so(p+1,q+1)$-invariant bilinear differential operators from ${\cal F}_\lambda\otimes{\cal F}_\mu$ to~${\cal F}_\nu$. The classification of linear $so(p+1,q+1)$-invariant differential operators from ${\cal F}_\lambda$ to ${\cal F}_\mu$ already known in the literature is obtained in a different manner. | Conformally invariant differential operators on tensor densities | 14,467 |
We use the one parameter fixed point theory of Geoghegan and Nicas to get information about the closed orbit structure of transverse gradient flows of closed 1-forms on a closed manifold M. We define a noncommutative zeta function in an object related to the first Hochschild homology group of the Novikov ring associated to the 1-form and relate it to the torsion of a natural chain homotopy equivalence between the Novikov complex and a simplicial complex of the universal cover of M. | One parameter fixed point theory and gradient flows of closed 1-forms | 14,468 |
We study the geometry of families of hypersurfaces in Eguchi-Hanson space that arise as complex line bundles over curves in $S^2$ and are three-dimensional, non-compact Riemannian manifolds, which are foliated in Hopf tori for closed curves. They are negatively curved, asymptotically flat spaces, and we compute the complete three-dimensional curvature tensor as well as the second fundamental form, giving also some results concerning their geodesic flow. We show the non-existence of $\L^p$-harmonic functions on these hypersurfaces for every $p \geq 1$ and arbitrary curves, and determine the infima of the essential spectra of the Laplace and of the square of the Dirac operator in the case of closed curves. For circles we also compute the $\L^2$-kernel of the Dirac operator in the sense of spectral theory and show that it is infinite dimensional. We consider further the Einstein Dirac system on these spaces and construct explicit examples of T-Killing spinors on them. | Geometric and analytic properties of families of hypersurfaces in
Eguchi-Hanson space | 14,469 |
We show that the indices of certain twisted Dirac operators vanish on a $Spin$-manifold $M$ of positive sectional curvature if the symmetry rank of $M$ is $\geq 2$ or if the symmetry rank is one and $M$ is two connected. We also give examples of simply connected manifolds of positive Ricci curvature which do not admit a metric of positive sectional curvature and positive symmetry rank. | Obstructions to positive curvature and symmetry | 14,470 |
Let $M$ be a closed connected manifold, $f$ be a Morse map from $M$ to a circle, $v$ be a gradient-like vector field satisfying the transversality condition. The Novikov construction associates to these data a chain complex $C_*=C_*(f,v)$. There is a chain homotopy equivalence between $C_*$ and completed simplicial chain complex of the corresponding infinite cyclic covering of $M$. The first main result of the paper is the construction of a functorial chain homotopy equivalence between these two complexes. The second main result states that the torsion of this chain homotopy equivalence equals to the Lefschetz zeta function of the gradient flow, if $v$ has only hyperbolic closed orbits. | Counting closed orbits of gradients of circle-valued maps | 14,471 |
We study two kinds of transformation groups of a compact locally conformally Kahler (l.c.K.) manifold. First we study compact l.c.K. manifolds with parallel Lee form by means of the existence of a holomorphic l.c.K. flow. Next, we introduce the Lee-Cauchy-Riemann (LCR) transformations as a class of diffeomorphisms preserving the specific G-structure of l.c.K. manifolds. We show that compact l.c.K. manifolds admitting a non-compact CC^* flow of LCR transformations are rigid: it is holomorphically conformal to a Hopf manifold with parallel Lee form. | Geometric flow on compact locally conformally Kahler manifolds | 14,472 |
This is a survey on quaternion Hermitian Weyl (locally conformally quaternion K\"ahler) and hyperhermitian Weyl (locally conformally hyperk\"ahler) manifolds. These geometries appear by requesting the compatibility of some quaternion Hermitian or hyperhermitian structure with a Weyl structure. The motivation for such a study is two-fold: it comes from the constantly growing interest in Weyl (and Einstein-Weyl) geometry and, on the other hand, from the necessity of understanding the existing classes of quaternion Hermitian manifolds. | Weyl structures on quaternionic manifolds. A state of the art | 14,473 |
We prove new lower bounds for the first eigenvalue of the Dirac operator on compact manifolds whose Weyl tensor or curvature tensor, respectively, is divergence free. In the special case of Einstein manifolds, we obtain estimates depending on the Weyl tensor. | Eigenvalue estimates for the Dirac operator depending on the Weyl
curvature tensor | 14,474 |
We shall give an axiomatic construction of Wess-Zumino-Witten actions valued in (G=SU(N)), (N\geq 3). It is realized as a functor ({WZ}) from the category of conformally flat four-dimensional manifolds to the category of line bundles with connection that satisfies, besides the axioms of a topological field theory, the axioms which abstract Wess-Zumino-Witten actions. To each conformally flat four-dimensional manifold (\Sigma) with boundary (\Gamma=\partial\Sigma), a line bundle (L=WZ(\Gamma)) with connection over the space (\Gamma G) of mappings from (\Gamma) to (G) is associated. The Wess-Zumino-Witten action is a non-vanishing horizontal section (WZ(\Sigma)) of the pull back bundle (r^{\ast}L) over (\Sigma G) by the boundary restriction (r). (WZ(\Sigma)) is required to satisfy a generalized Polyakov-Wiegmann formula with respect to the pointwise multiplication of the fields (\Sigma G). Associated to the WZW-action there is a geometric descrption of extensions of the Lie group (\Omega^3G) due to J. Mickelsson. In fact we shall construct two abelian extensions of (\Omega^3G) that are in duality. | Four-dimensional Wess-Zumino-Witten actions | 14,475 |
This is primarily a survey of the developments in the theory of harmonic maps of finite uniton number (or unitons) which have taken place since the introduction of extended solutions by Uhlenbeck. Such maps include all harmonic maps from the two-sphere to a compact Lie group or symmetric space. Extended solutions are equivalent to harmonic maps, but the advantage of extended solutions is the fact that they are solutions to a zero curvature equation. The closely related complex extended solutions (or complex extended frames) are in fact even more advantageous, as has been pointed out by Dorfmeister, Pedit and Wu. We use them to review the existing theory and also to prove some new results concerning harmonic maps into the unitary group. | An update on harmonic maps of finite uniton number, via the zero
curvature equation | 14,476 |
We give an explicit formula for the projectively invariant quantization map between the space of symbols of degree three and the space of third-order linear differential operators, both viewed as modules over the group of diffeomorphisms and the Lie algebra of vector fields on a manifold. | Fromula for the Projectively Invariant Quantization on Degree Three | 14,477 |
We define the pull-back of a smooth principal fibre bundle, and show that it has a natural principal fibre bundle structure. Next, we analyse the relationship between pull-backs by homotopy equivalent maps. The main result of this article is to show that for a principal fibre bundle over a paracompact manifold, there is a principal fibre bundle isomorphism between pull-backs obtained from homotopic maps. This enables simple proofs of several results on the structure of principal fibre bundles. No new results are obtained---this is simply an accessible presentation of an important idea. | An Introduction to Pull-backs of Bundles and Homotopy Invariance | 14,478 |
We study affine maps between affine manifolds. Even when the fibers are compact and diffeomorphic, two of them can inherit different affine structures from the source space. This leads to a fixed linear holonomy deformation theory of the affine structure of an affine manifold. We found various conditions which make the fibers to be affinely isomorphic. We also classify affine bundles which total space is a compact and complete affine manifold. | Fibres affines | 14,479 |
We systematically develop a transform of the Fourier-Mukai type for sheaves on symplectic manifolds $X$ of any dimension fibred in Lagrangian tori. One obtains a bijective correspondence between unitary local systems supported on Lagrangian submanifolds of $X$ and holomorphic vector bundles with compatible unitary connections supported on complex submanifolds of the relative Jacobian of $X$ (suitable conditions being verified on both sides). | A Fourier transform for sheaves on Lagrangian families of real tori | 14,480 |
In this paper we study n-composition series of affine manifolds. One composition series are classified using gerbe theory. It is natural to think that n-composition series must be classified using n-gerbe theory. In the last section of this, we propose a notion of abelian n-gerbe theory | Composition series of affine manifolds and n-gerbes | 14,481 |
We call a quaternionic Kaehler manifold with non-zero scalar curvature, whose quaternionic structure is trivialized by a hypercomplex structure, a hyper-Hermitian quaternionic Kaehler manifold. We prove that every locally symmetric hyper-Hermitian quaternionic Kaehler manifold is locally isometric to the quaternionic projective space or to the quaternionic hyperbolic space. We describe locally the hyper-Hermitian quaternionic Kaehler manifolds with closed Lee form and show that the only complete simply connected such manifold is the quaternionic hyperbolic space. | Hyper-Hermitian quaternionic Kaehler manifolds | 14,482 |
We prove the Goldman-Parker Conjecture: A complex hyperbolic ideal triangle group is directly embedded in PU(2,1) if and only if the product of its three standard generators is not elliptic. We also prove that such a group is indiscrete if the product of its three standard generators is elliptic. A novel feature of this paper is that it uses a rigorous computer assisted proof to deal with difficult geometric estimates. | Ideal triangle groups, dented tori, and numerical analysis | 14,483 |
In this paper we shall establish that properly embedded constant mean curvature one surfaces in H^3 of finite topology are of finite total curvature and each end is regular. In particular, this implies the horosphere is the only simply connected such example, and the catenoid cousins the only annular examples of this nature. In general each annular end of such a surface is asymptotic to an end of a horosphere or an end of a catenoid cousin. | The geometry of finite topology Bryant surfaces | 14,484 |
We make the category BGrb_M of bundle gerbes on a manifold M into a 2-category by providing 2-cells in the form of transformations of bundle gerbe morphisms. This description of BGrb_M as a 2-category is used to define the notion of a bundle 2-gerbe. To every bundle 2-gerbe on M is associated a class in H^4(M;Z). We define the notion of a bundle 2-gerbe connection and show how this leads to a closed, integral differential 4-form on M which represents the image in real cohomology of the class in H^4(M;Z). Some examples of bundle 2-gerbes are discussed, including the bundle 2-gerbe associated to a principal G-bundle P \to M. It is shown that the class in H^4(M;Z) associated to this bundle 2-gerbe coincides with the first Pontryagin class of P --- this example was previously considered from the point of view of 2-gerbes by Brylinski and McLaughlin. | Bundle 2-gerbes | 14,485 |
We propose a geometric correspondence between (a) linearly degenerate systems of conservation laws with rectilinear rarefaction curves and (b) congruences of lines in projective space whose developable surfaces are planar pencils of lines. We prove that in projective 4-space such congruences are necessarily linear. Based on the results of Castelnuovo, the classification of three-component systems is obtained, revealing a close relationship of the problem with projective geometry of the Veronese variety and the theory of associativity equations of two-dimensional topological field theory. | Systems of conservation laws of Temple class, equations of associativity
and linear congruences in projective space | 14,486 |
For a real valued periodic smooth function u on R, $n\ge 0$, one defines the osculating polynomial $\phi_s$ (of order 2n+1) at a point $s\in R$ to be the unique trigonometric polynomial of degree n, whose value and first 2n derivatives at s coincide with those of u at s. We will say that a point s is a clean maximal flex (resp. clean minimal flex) of the function u on $S^1$ if and only if $\phi_s\ge u$ (resp. $\phi_s\le u$) and the preimage $(\phi-u)^{-1}(0)$ is connected. We prove that any smooth periodic function u has at least n+1 clean maximal flexes of order 2n+1 and at least n+1 clean minimal flexes of order 2n+1. The assertion is clearly reminiscent of Morse theory and generalizes the classical four vertex theorem for convex plane curves. | A global theory of flexes of periodic functions | 14,487 |
We prove that, from an Einstein manifold of dimension greater than or equal to five, there are just two types of harmonic morphism with one-dimensional fibres. This generalizes a result of R.L. Bryant who obtained the same conclusion under the assumption that the domain has constant curvature. | Harmonic morphisms with one-dimensional fibres on Einstein manifolds | 14,488 |
Any Kaehler metric on the ball which is strongly asymptotic to complex hyperbolic space and whose scalar curvature is no less than the one of the complex hyperbolic space must be isometrically biholomorphic to it. This result has been known for some time in odd complex dimension and we provide here a proof in even dimension. | Rigidity at infinity for even-dimensional asymptotically complex
hyperbolic spaces | 14,489 |
Article is devoted to the Examples 2 and 3 of the symplectic solvable Lie groups $R$ with some special cohomological properties, which have been constructed by Benson and Gordon. But they are not succeeded in constructing corresponding compact forms for symplectic structures on these Lie groups. Recently A.Tralle proved that there is no compact form in the Example 3. But his proof is rather complicated and uses some very special topological methods. We propose much more simpler (and purely algebraic) method to prove the main result of the Tralle's paper. Moreover we prove that for Example 2 there is no compact form too. But it appears that some modification of the construction of the Example 2 gives some other example of a solvable Lie group $R^{\prime}$ with the same cohomological properties as $R$, but with a compact form. | On some examples in Symplectic Topology | 14,490 |
The Levi-Civita connection and geodesic equations for a stationary spacetime are studied in depth. General formulae which generalize those for warped products are obtained. These results are applicated to some regions of Kerr spacetime previously studied by using variational methods. We show that they are neither space-convex nor geodesically connected. Moreover, the whole stationary part of Kerr spacetime is not geodesically connected, except when the angular momentum is equal to zero (Schwarzschild spacetime). | Geodesics in stationary spacetimes. Application to Kerr spacetime | 14,491 |
In the first part of this paper, given a smooth family of Dirac-type operators on an odd-dimensional closed manifold, we construct an abelian gerbe-with-connection whose curvature is the three-form component of the Atiyah-Singer families index theorem. In the second part of the paper, given a smooth family of Dirac-type operators whose index lies in the i-th filtration of the reduced K-theory of the parametrizing space, we construct a set of Deligne cohomology class of degree i whose curvatures are the i-form component of the Atiyah-Singer families index theorem. | Higher-Degree Analogs of the Determinant Line Bundle | 14,492 |
We obtain sufficient conditions exlcuding the existence of non-trivial distribution sections of bundles over the boundary of symmetric spaces of negative curvature which are invariant with respect to a geometrically finite group of isometries and are supported on the limit set in a strong sense. | Nonexistence of invariant distributions supported on the limit set | 14,493 |
Let G be a Lie groupoid over M such that the target-source map from G to M x M is proper. We show that, if O is an orbit of finite type (i.e. which admits a proper function with finitely many critical points), then the restriction G|U of G to some neighborhood U of O in M is isomorphic to a similar restriction of the action groupoid for the linear action of the transitive groupoid G|O on the normal bundle NO. The proof uses a deformation argument based on a cohomology vanishing theorem, along with a slice theorem which is derived from a new result on submersions with a fibre of finite type. | Linearization of Regular Proper Groupoids | 14,494 |
We prove that there are no pseudoholomorphic theories of anything other than curves, even if one allows more general spaces than almost complex manifolds. The proof is elementary, except for theories of pseudoholomorphic hypersurfaces, where topological techniques are needed. Surprisingly, hypersurface theories exist ``microlocally'' (in great abundance) to all orders perturbatively, but not ``locally.'' | Analogues of Complex Geometry | 14,495 |
The problem of immersing a simply connected surface with a prescribed shape operator is discussed. From classical and more recent work, it is known that, aside from some special degenerate cases, such as when the shape operator can be realized by a surface with one family of principal curves being geodesic, the space of such realizations is a convex set in an affine space of dimension at most 3. The cases where this maximum dimension of realizability is achieved have been classified and it is known that there are two such families of shape operators, one depending essentially on three arbitrary functions of one variable (called Type I in this article) and another depending essentially on two arbitrary functions of one variable (called Type II in this article). In this article, these classification results are rederived, with an emphasis on explicit computability of the space of solutions. It is shown that, for operators of either type, their realizations by immersions can be computed by quadrature. Moreover, explicit normal forms for each can be computed by quadrature together with, in the case of Type I, by solving a single linear second order ODE in one variable. (Even this last step can be avoided in most Type I cases.) The space of realizations is discussed in each case, along with some of their remarkable geometric properties. Several explicit examples are constructed (mostly already in the literature) and used to illustrate various features of the problem. | On surfaces with prescribed shape operator | 14,496 |
We show that a totally geodesic submanifold of a symmetric space satisfying certain conditions admits an extension to a minimal submanifold of dimension one higher, and we apply this result to construct new examples of complete embedded minimal submanifolds in simply connected noncompact globally symmetric spaces. | A class of complete embedded minimal submanifolds in noncompact
symmetric spaces | 14,497 |
We show how certain diffeomorphism-invariant functionals on differential forms in dimensions 6,7 and 8 generate in a natural way special geometrical structures in these dimensions: metrics of holonomy G2 and Spin(7), metrics with weak holonomy SU(3) and G2, and a new and unexplored example in dimension 8. The general formalism becomes a practical tool for calculating homogeneous or cohomogeneity one examples, and we illustrate this with some newly discovered cases of Spin(7) and G2 metrics. | Stable forms and special metrics | 14,498 |
The class of surfaces in 3-space possessing nontrivial deformations which preserve principal directions and principal curvatures (or, equivalently, the shape operator) was investigated by Finikov and Gambier as far back as in 1933. We review some of the known examples and results, demonstrate the integrability of the corresponding Gauss-Codazzi equations and draw parallels between this geometrical problem and the theory of compatible Poisson brackets of hydrodynamic type. It turns out that coordinate hypersurfaces of the n-orthogonal systems arising in the theory of compatible Poisson brackets of hydrodynamic type must necessarily possess deformations preserving the shape operator. | Surfaces in 3-space possessing nontrivial deformations which preserve
the shape operator | 14,499 |
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