Showing papers in "Differential Geometry and Its Applications in 2003"
TL;DR: In this article, the authors studied the interplay between invariant-theoretic and Riemannian aspects of real Lie algebras and determined all the Lie groups which can be endowed with only one left invariant metric up to isometry and scaling.
Abstract: Each point of the variety of real Lie algebras is naturally identified with a left invariant Riemannian metric on a Lie group. We study the interplay between invariant-theoretic and Riemannian aspects of this variety. In particular, using the special critical point behavior of certain natural functional on the variety, we determine all the Lie groups which can be endowed with only one left invariant metric up to isometry and scaling, proving first that they correspond to Lie algebras whose only degeneration is to the abelian one. We also find all the Lie algebras which degenerate to the Lie algebra of the hyperbolic space, and all the possible degenerations for 3-dimensional real Lie algebras, by using well known descriptions of left invariant metrics satisfying some pinching curvature conditions. Finally, as another interaction, the closed S L ( n )-orbits on the variety are classified, and explicit curves of Einstein solvmanifolds are provided by using curves of closed orbits of the representation Λ 2 S L ( m )⊗S L ( n ).
90 citations
TL;DR: In this paper, the authors used a variational approach to determine all the 7-dimensional rank-one Einstein solvmanifolds of Lie algebras of dimension 6.
Abstract: We use a variational approach to determine explicitly all the 7-dimensional rank-one Einstein solvmanifolds. We prove that each one of the 34 nilpotent Lie algebras of dimension 6 admits a rank-one solvable extension which can be endowed with an Einstein left-invariant Riemannian metric. This also produces 34 Ricci soliton metrics on R 6 which are homogeneous.
77 citations
TL;DR: In this article, a lower bound on the first positive eigenvalue of the Dirac operator on a compact spin manifold was proved, depending only on the spin structure and the conformal type.
Abstract: Let D be the Dirac operator on a compact spin manifold M . Assume that 0 is in the spectrum of D . We prove the existence of a lower bound on the first positive eigenvalue of D depending only on the spin structure and the conformal type.
57 citations
TL;DR: In this article, the authors developed a theory of conformal Killing tensors and proved that the Hamilton-Jacobi equation for Riemannian manifold admits a conformal tensor whose torsion vanishes, and whose eigenfunctions are independent.
Abstract: If a Riemannian manifold admits a conformal Killing tensor whose torsion, in the sense of Nijenhuis, vanishes, and whose eigenfunctions are independent, then the Hamilton–Jacobi equation for its geodesics is solvable by separation of variables The paper is devoted to developing a theory of this class of conformal Killing tensors, including an explanation of this result
42 citations
TL;DR: In this article, it was shown that sufficiently close inner parallel sets and closures of the complements of outer parallel sets to a d-dimensional Lipschitz manifold in R d with boundary have locally positive reach and the normal cycle of the Lipschnitz manifold can be defined as limit of normal cycles of the parallel sets in the flat seminorms for currents.
Abstract: It is shown that sufficiently close inner parallel sets and closures of the complements of outer parallel sets to a d-dimensional Lipschitz manifold in R d with boundary have locally positive reach and the normal cycle of the Lipschitz manifold can be defined as limit of normal cycles of the parallel sets in the flat seminorms for currents, provided that the normal cycles of the parallel set have locally bounded mass. The Gauss–Bonnet formula and principal kinematic formula are proved for these normal cycles. It is shown that locally finite unions of non-osculating sets with positive reach of full dimension, as well as the closures of their complements, admit such a definition of normal cycle.
41 citations
TL;DR: In this paper, a generalised notion of connection on a fiber bundle E over a manifold M is presented, characterised by a smooth distribution on E which projects onto a (not necessarily integrable) distribution on M and which, in addition, is parametrised in some specific way by a vector bundle map from a prescribed vector bundle over M into TM.
Abstract: A generalised notion of connection on a fibre bundle E over a manifold M is presented. These connections are characterised by a smooth distribution on E which projects onto a (not necessarily integrable) distribution on M and which, in addition, is ‘parametrised’ in some specific way by a vector bundle map from a prescribed vector bundle over M into TM. Some basic properties of these generalised connections are investigated. Special attention is paid to the class of linear connections over a vector bundle map. It is pointed out that not only the more familiar types of connections encountered in the literature, but also the recently studied Lie algebroid connections, can be recovered as special cases within this more general framework.
41 citations
TL;DR: In this paper, the geometry of doubly extended Lie groups with their natural bi-invariant metric is studied. But the authors focus on the space of parallel spinors and not on the curvature of the groups.
Abstract: In the present paper we study the geometry of doubly extended Lie groups with their natural biinvariant metric. We describe the curvature, the holonomy and the space of parallel spinors. This is completely done for all simply connected groups with biinvariant metric of Lorentzian signature (1, n −1), of signature (2, n −2) and of signature ( p , q ), where p + q ⩽6. Furthermore, some special series with higher signature are discussed.
34 citations
TL;DR: In this paper, the authors provide a geometrically oriented approach to the theory of orbispaces originally introduced by G.Schwarz and W.Chen and show that every symplectic orbispace has a deformation quantization.
Abstract: In the first part of this article we provide a geometrically oriented approach to the theory of orbispaces originally introduced by G. Schwarz and W. Chen. We explain the notion of a vector orbibundle and characterize the good sections of a reduced vector orbibundle as the smooth stratified sections. In the second part of the article we elaborate on the quantizability of a symplectic orbispace. By adapting Fedosov's method to the orbispace setting we show that every symplectic orbispace has a deformation quantization. As a byproduct we obtain that every symplectic orbifold possesses a star product.
33 citations
TL;DR: In this article, the curvature of the twistor bundle of an even-dimensional Riemannian manifold M whose sections are the almost-Hermitian structures of M was studied.
Abstract: We show how the equations for harmonic maps into homogeneous spaces generalize to harmonic sections of homogeneous fibre bundles. Surprisingly, the generalization does not explicitly involve the curvature of the bundle. However, a number of special cases of the harmonic section equations (including the new condition of super-flatness) are studied in which the bundle curvature does appear. Some examples are given to illustrate these special cases in the non-flat environment. The bundle in question is the twistor bundle of an even-dimensional Riemannian manifold M whose sections are the almost-Hermitian structures of M.
31 citations
TL;DR: In this paper, the authors classify the total spaces of bundles over the four sphere with fiber a three sphere up to orientation preserving homotopy equivalence, homeomorphism and diffeomorphism.
Abstract: We classify the total spaces of bundles over the four sphere with fiber a three sphere up to orientation preserving and reversing homotopy equivalence, homeomorphism and diffeomorphism. These total spaces have been of interest to both topologists and geometers. It has recently been shown by Grove and Ziller (Ann. of Math. (2) 152 (2000) 331–367) that each of these total spaces admits metrics with nonnegative sectional curvature.
29 citations
TL;DR: In this article, a partial classification of homogeneous Einstein 6-manifolds with positive scalar curvature is presented. But this classification is restricted to the case of homogenous 6-Manifolds.
Abstract: This paper is devoted to the partial classification of homogeneous Einstein 6-manifolds with positive scalar curvature.
TL;DR: In this paper, it was shown that pointwise Osserman manifolds are two-point homogeneous, provided 8∆n and n≠2, 4, respectively, under the assumption that the multiplicities of the eigenvalues are not 7 and 8.
Abstract: Let Mn be a Riemannian manifold. For a point p∈Mn and a unit vector X∈TpMn, the Jacobi operator is defined by R X =R(X,· )X , where R is the curvature tensor. The manifold Mn is called pointwise Osserman if, for every p∈Mn, the spectrum of the Jacobi operator does not depend of the choice of X, and is called globally Osserman if it depends neither of X, nor of p. R. Osserman conjectured that globally Osserman manifolds are two-point homogeneous. We prove the following: (1) A pointwise Osserman manifold Mn is two-point homogeneous, provided 8∤n and n≠2, 4; a globally Osserman manifold Mn is two-point homogeneous, provided 8∤n ; (2) Let Mn be a globally Osserman manifold with the Jacobi operator having exactly two eigenvalues. In the case n=16, assume that the multiplicities of the eigenvalues are not 7 and 8, respectively. Then Mn is two-point homogeneous.
TL;DR: In this article, the Schouten-Nijenhuis bracket of covariant symmetric tensor fields defined by the cotangent Lie algebroid of a manifold M to its tangent bundle TM is interpreted as a complete lift of a Poisson structure from a manifold P to TM.
Abstract: The paper starts with an interpretation of the complete lift of a Poisson structure from a manifold M to its tangent bundle TM by means of the Schouten–Nijenhuis bracket of covariant symmetric tensor fields defined by the cotangent Lie algebroid of M. Then, we discuss Poisson structures of TM which have a graded restriction to the fiberwise polynomial algebra; they must be π-related ( π :TM→M ) with a Poisson structure on M. Furthermore, we define transversal Poisson structures of a foliation, and discuss bivector fields of TM which produce graded brackets on the fiberwise polynomial algebra, and are transversal Poisson structures of the foliation by fibers. Finally, such bivector fields are produced by a process of horizontal lifting of Poisson structures from M to TM via connections.
TL;DR: In this paper, the inverse problem of Lagrangian dynamics is solved for the geodesic spray associated to the canonical symmetric linear connection on a Lie group of dimension three or less.
Abstract: The inverse problem of Lagrangian dynamics is solved for the geodesic spray associated to the canonical symmetric linear connection on a Lie group of dimension three or less. The degree of generality is obtained in each case and concrete Lagrangians are written down.
TL;DR: In this paper, the authors study timelike surfaces in Lorentzian space forms which admit a one-parameter family of isometric deformations preserving the mean curvature.
Abstract: We study timelike surfaces in Lorentzian space forms which admit a one-parameter family of isometric deformations preserving the mean curvature.
TL;DR: In this paper, the Hopf Index Theorem for Riemannian foliations has been proved for a closed manifold with non-degenerate basic vector fields at critical leaf closures.
Abstract: We formulate and prove an analog of the Hopf Index Theorem for Riemannian foliations. We compute the basic Euler characteristic of a closed Riemannian manifold as a sum of indices of a non-degenerate basic vector field at critical leaf closures. The primary tool used to establish this result is an adaptation to foliations of the Witten deformation method.
TL;DR: In this article, it was shown that #8(S 2 × S 3 ) admits two 8-dimensional complex families of inequivalent non-regular Sasakian-Einstein structures.
Abstract: We show that #8( S 2 × S 3 ) admits two 8-dimensional complex families of inequivalent non-regular Sasakian–Einstein structures. These are the first known non-regular Sasakian–Einstein metrics on this 5-manifold.
TL;DR: In this paper, the authors established several sufficient conditions for a compact spacelike surface with non-degenerate second fundamental form in the 3-dimensional de Sitter space to be spherical.
Abstract: In this paper, we establish several sufficient conditions for a compact spacelike surface with non-degenerate second fundamental form in the 3-dimensional de Sitter space to be spherical. With this aim, we develop a formula for these surfaces which involves the mean and Gaussian curvatures of the first fundamental form and the Gaussian curvature of the second fundamental form. By means of that formula, we prove, for instance, that the totally umbilical round spheres are the only compact spacelike surfaces such that the second fundamental form is nondegenerate and has constant Gaussian curvature. 2003 Elsevier Science B.V. All rights reserved.
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TL;DR: Ortega et al. as mentioned in this paper generalize the notions of dual pair and polarity introduced by S. Lie and A. Weinstein in order to accommodate very relevant situations where the application of these ideas is desirable.
Abstract: We generalize the notions of dual pair and polarity introduced by S. Lie (1890) and A. Weinstein (1983) in order to accommodate very relevant situations where the application of these ideas is desirable. The new notion of polarity is designed to deal with the loss of smoothness caused by the presence of singularities that are encountered in many problems in Poisson and symplectic geometry. We study in detail the relation between the newly introduced dual pairs, the quantum notion of Howe pair, and the symplectic leaf correspondence of Poisson manifolds in duality. The dual pairs arising in the context of symmetric Poisson manifolds are treated with special attention. We show that in this case and under very reasonable hypotheses we obtain a particularly well behaved kind of dual pairs that we call von Neumann pairs. Some of the ideas that we present in this paper shed some light on the optimal momentum maps introduced in [J.-P. Ortega, T.S. Ratiu, The optimal momentum map, in: P. Holmes, P. Newton, A. Weinstein (Eds.), Geometry, Dynamics and Mechanics: 60th Birthday Volume for J.E. Marsden, Springer-Verlag, New York, 2002, pp. 319–362].
TL;DR: Several new invariants of Lie algebroids have been discovered recently as discussed by the authors, and an overview of these invariants and relationships between them can be found in Section 2.1.
Abstract: Several new invariants of Lie algebroids have been discovered recently. We give an overview of these invariants and establish several relationships between them.
TL;DR: In this paper, the vanishing of the first Betti number on compact Hermitian-Weyl manifold admits a Weyl structure whose Ricci tensor satisfies certain positivity conditions, thus obtaining a Bochner-type vanishing theorem in Weyl geometry.
Abstract: 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.
TL;DR: In this paper, the authors introduced the notion of total curvature of curves (which agrees with the usual one in the piecewise smooth case) in spaces of Alexandrov curvature bounded above.
Abstract: We introduce the notion of total curvature of curves (which agrees with the usual one in the piecewise smooth case) in spaces of Alexandrov curvature bounded above. Basic properties of total curvature, including rectifiability of curves of finite total curvature and additivity of total curvature, are then obtained. A sharp upper estimate of a type due to Schmidt on the length of a curve in a CAT( K ) space is also given in terms of its total curvature and the distance between its endpoints.
TL;DR: In this paper, a family of Cayley transforms of a homogeneous Siegel domain D is introduced, which maps D biholomorphically onto a bounded domain in a complex Euclidean vector space.
Abstract: In this paper we introduce a family of Cayley transforms of a homogeneous Siegel domain D. Each of these transforms is birational and maps D biholomorphically onto a bounded domain in a complex Euclidean vector space.
TL;DR: In this article, the authors complete the list of normal forms for effective 3-forms with constant coefficients with respect to the natural action of symplectomorphisms in R 6, and show that the 3-form which corresponds to the Special Lagrangian equation is among the new members of the classification.
Abstract: We complete the list of normal forms for effective 3-forms with constant coefficients with respect to the natural action of symplectomorphisms in R 6 . We show that the 3-form which corresponds to the Special Lagrangian equation is among the new members of the classification. The symplectic symmetry algebras and their Cartan prolongations for these forms are computed and a local classification theorem for the corresponding Monge–Ampere equations is proved.
TL;DR: In this article, the authors deal with Radon transforms for generalized flag manifolds in the framework of quasi-equivariant D -modules and give a sufficient condition in order that a D -module associated to an equivariant locally free O -module is transformed into an object of the same type.
Abstract: In this paper we deal with Radon transforms for generalized flag manifolds in the framework of quasi-equivariant D -modules. We shall follow the method employed by Baston–Eastwood and analyze the Radon transform using the Bernstein–Gelfand–Gelfand resolution and the Borel–Weil–Bott theorem. We shall determine the transform completely on the level of the Grothendieck groups. Moreover, we point out a vanishing criterion and give a sufficient condition in order that a D -module associated to an equivariant locally free O -module is transformed into an object of the same type. The case of maximal parabolic subgroups is studied in detail.
TL;DR: In this paper, it is shown that there are alternative Riemannian structures on the space of curves that yield flows that preserve symmetries, including rotation symmetry and symmetry of reflection.
Abstract: The ‘traditional’ curve-straightening flow is based on one of the standard Sobolev inner products and it is known to break certain symmetries of reflection. The purpose of this paper is to show that there are alternative Riemannian structures on the space of curves that yield flows that preserve symmetries. This feature comes at a price. In one symmetrizing metric the gradient vector fields are considerably more demanding to compute. In another symmetrizing metric smoothness is lost. This investigation will also explain the phenomena of ‘spinning’ as observed in several examples in the traditional flow. Three classes of alternative Riemannian structures are examined. The first class includes the traditional metric as a special case and is shown to never preserve both rotation symmetries and symmetries of reflection. The second class consists of a single metric corresponding to one of the standard Sobolev metrics, and is shown to preserve both types of symmetries. The third class also includes the traditional metric but it is shown that there is a unique different metric in this class, which preserves both types of symmetries. This particular metric generally yields smooth vector fields, which when evaluated at a smooth function do not give a smooth element of the corresponding tangent space. The third class is nevertheless ‘preferred’ since it has the distinction that it ‘respects’ the projection induced by the derivative operator onto the tangent bundle of the space of derivatives. The paper concludes with a number of graphical illustrations that show preserved symmetry and removal of spinning.
TL;DR: In this article, the singular Lagrangian foliation of a 2-degree of freedom integrable Hamiltonian system is shown to be symplectically equivalent to the linearized foliation in a neighbourhood of a non-degenerate singular orbit.
Abstract: We prove that the singular Lagrangian foliation of a 2-degree of freedom integrable Hamiltonian system, is symplectically equivalent to the linearized foliation in a neighbourhood of a non-degenerate singular orbit.
TL;DR: In this article, it was shown that the Lax pairs of the self-dual Yang-Mills equations and their reductions are of this type and the bracket on the cochains induced by commutation of symmetries.
Abstract: We study the equation $\mathcal{E}_{\mathrm{fc}}$ of flat connections in a given fiber bundle and discover a specific geometric structure on it, which we call a \emph{flat representation}. We generalize this notion to arbitrary PDE and prove that flat representations of an equation $\mathcal{E}$ are in $1$-$1$ correspondence with morphisms $\varphi\colon\mathcal{E}\to\mathcal{E}_{\mathrm{fc}}$, where $\mathcal{E}$ and $\mathcal{E}_{\mathrm{fc}}$ are treated as submanifolds of infinite jet spaces. We show that flat representations include several known types of zero-\hspace{0pt} curvature formulations of PDEs. In particular, the Lax pairs of the self-dual Yang--Mills equations and their reductions are of this type. With each flat representation $\varphi$ we associate a complex $C_\varphi$ of vector-\hspace{0pt} valued differential forms such that $H^1(C_\varphi)$ describes infinitesimal deformations of the flat structure, which are responsible, in particular, for parameters in B\"{a}cklund transformations. In addition, each higher infinitesimal symmetry $S$ of $\mathcal{E}$ defines a $1$-\hspace{0pt} cocycle $c_S$ of $C_\varphi$. Symmetries with exact $c_S$ form a subalgebra reflecting some geometric properties of $\mathcal{E}$ and $\varphi$. We show that the complex corresponding to $\mathcal{E}_{\mathrm{fc}}$ itself is $0$-\hspace{0pt} acyclic and $1$-\hspace{0pt} acyclic (independently of the bundle topology), which means that higher symmetries of $\mathcal{E}_{\mathrm{fc}}$ are exhausted by generalized gauge ones, and compute the bracket on $0$-\hspace{0pt} cochains induced by commutation of symmetries.
TL;DR: In this paper, the universal covering of a compact Kahler manifold with a residually solvable fundamental group is shown to be holomorphically convex, and a factorization theorem for such representations is derived.
Abstract: In the paper we study solvable matrix representations of fundamental groups of compact Kahler manifolds (Kahler groups). One of our main results is a factorization theorem for such representations. Further, we explore the structure of certain Kahler groups. As an application we prove that the universal covering of a compact Kahler manifold with a residually solvable fundamental group is holomorphically convex.
TL;DR: In this paper, all first order Lagrangian densities on the bundle of connections associated to P that are invariant under the Lie algebra of infinitesimal automorphisms are shown to be variationally trivial and to give constant actions that equal the characteristic numbers of P if dimM is even and zero ifdimM is odd.
Abstract: Given a principal bundle P→M we classify all first order Lagrangian densities on the bundle of connections associated to P that are invariant under the Lie algebra of infinitesimal automorphisms. These are shown to be variationally trivial and to give constant actions that equal the characteristic numbers of P if dimM is even and zero if dimM is odd. In addition, we show that variationally trivial Lagrangians are characterized by the de Rham cohomology of the base manifold M and the characteristic classes of P of arbitrary degree