Showing papers in "Differential Geometry and Its Applications in 2008"

Totally geodesic submanifolds of the complex quadric

Abstract: In this article, relations between the root space decomposition of a Riemannian symmetric space of compact type and the root space decompositions of its totally geodesic submanifolds (symmetric subspaces) are described. These relations provide an approach to the classification of totally geodesic submanifolds in Riemannian symmetric spaces; this is exemplified by the classification of the totally geodesic submanifolds in the complex quadric Q m : = SO ( m + 2 ) / ( SO ( 2 ) × SO ( m ) ) obtained in the second part of the article. The classification shows that the earlier classification of totally geodesic submanifolds of Q m by Chen and Nagano (see [B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977) 745–755]) is incomplete. More specifically, two types of totally geodesic submanifolds of Q m are missing from [B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces, I, Duke Math. J. 44 (1977) 745–755]: The first type is constituted by manifolds isometric to C P 1 × R P 1 ; their existence follows from the fact that Q 2 is (via the Segre embedding) holomorphically isometric to C P 1 × C P 1 . The second type consists of 2-spheres of radius 1 2 10 which are neither complex nor totally real in Q m .

Structures on generalized Sasakian-space-forms

Abstract: In this paper, contact metric and trans-Sasakian generalized Sasakian-space-forms are deeply studied. We present some general results for manifolds with dimension greater than or equal to 5, and we also pay a special attention to the 3-dimensional cases.

On δ-homogeneous Riemannian manifolds

Abstract: We study in this paper previously defined by V.N. Berestovskii and C.P. Plaut δ -homogeneous spaces in the case of Riemannian manifolds and prove that they constitute a new proper subclass of geodesic orbit (g.o.) spaces with non-negative sectional curvature, which properly includes the class of all normal homogeneous Riemannian spaces.

Left invariant contact structures on Lie groups

Abstract: Amongst other results, we perform a ‘contactization’ method to construct, in every odd dimension, many contact Lie groups with a discrete center, unlike the usual (classical) contactization which only produces Lie groups with a non-discrete center. We discuss some applications and consequences of such a construction, construct several examples and derive some properties. We give classification results in low dimensions. A complete list is supplied in dimension 5. In any odd dimension greater than 5, there are infinitely many locally non-isomorphic solvable contact Lie groups. We also characterize solvable contact Lie algebras whose derived ideal has codimension one. For simplicity, most of the results are given in the Lie algebra version.

The conformal Killing equation on forms; prolongations and applications

Abstract: We construct a conformally invariant vector bundle connection such that its equation of parallel transport is a first order system that gives a prolongation of the conformal Killing equation on differential forms. Parallel sections of this connection are related bijectively to solutions of the conformal Killing equation. We construct other conformally invariant connections, also giving prolongations of the conformal Killing equation, that bijectively relate solutions of the conformal Killing equation on k forms to a twisting of the conformal Killing equation on k' forms for various integers k'. These tools are used to develop a helicity raising and lowering construction in the general setting and on conformally Einstein manifolds.

New examples of entire maximal graphs in H2×R1

Abstract: In this paper we obtain new explicit examples of complete and non-complete entire maximal graphs in H 2 × R 1 . The existence of these entire maximal graphs shows that entire maximal graphs in this Lorentzian product space are not necessarily complete, on the contrary that in the Lorentz–Minkowski space. Moreover, in [A.L. Albujer, L.J. Alias, Calabi–Bernstein results for maximal surfaces in Lorentzian product spaces, Preprint, 2006], the author jointly with Alias gave a Calabi–Bernstein theorem for maximal surfaces immersed into the Lorentzian product space M 2 × R 1 , where M 2 is a connected Riemannian surface of non-negative Gaussian curvature, and these examples show that the assumption on K M is necessary.

Rigidity theorems for complete spacelike hypersurfaces with constant scalar curvature in De Sitter space

Abstract: In this paper we give a partially affirmative answer to the following question posed by Haizhong Li: is a complete spacelike hypersurface in De Sitter space S 1 n + 1 ( c ) , n ⩾ 3 , with constant normalized scalar curvature R satisfying n − 2 n c ⩽ R ⩽ c totally umbilical?

Stability of spacelike hypersurfaces in foliated spacetimes

Abstract: Given a generalized M ¯ n + 1 = I × ϕ F n Robertson–Walker spacetime whose warping function verifies a certain convexity condition, we classify strongly stable spacelike hypersurfaces with constant mean curvature. More precisely, we will show that given x : M n → M ¯ n + 1 a closed, strongly stable spacelike hypersurface of M ¯ n + 1 with constant mean curvature H, if the warping function ϕ satisfying ϕ ″ ⩾ max { H ϕ ′ , 0 } along M, then M n is either maximal or a spacelike slice M t 0 = { t 0 } × F , for some t 0 ∈ I .

Real hypersurfaces in complex projective space with recurrent structure Jacobi operator

Abstract: We classify real hypersurfaces of complex projective space C P m , m ⩾ 3 , with D -recurrent structure Jacobi operator and apply this result to prove the non-existence of such hypersurfaces with recurrent structure Jacobi operator.

A classification of pseudo-Einstein hypersurfaces in CP2

Abstract: In this paper we present a classification of the pseudo-Einstein hypersurfaces M 3 in the complex space form C P 2 of constant holomorphic curvature 4 c = 4 r 2 . Specifically, (i) Such a hypersurface must be Hopf; (ii) In addition to the geodesic spheres, all tubes of radius π 4 r around holomorphic curves are pseudo-Einstein; (iii) All pseudo-Einstein hypersurfaces in C P 2 are generically (in a sense we will make precise) of this form; (iv) The only compact pseudo-Einstein hypersurfaces are the geodesic spheres.

Cartan connection associated with a subriemannian structure

Abstract: For each subriemannian manifold of constant subriemannian symbol we construct a Cartan connection canonically associated with this structure.

Biharmonic maps, conformal deformations and the Hopf maps

Abstract: We consider harmonic semi-conformal maps between two Riemannian manifolds. By deforming conformally the codomain metric, we construct new examples of non-harmonic biharmonic maps.

Spectral properties of bipolar minimal surfaces in S^4

Abstract: The i-th eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of fixed area. Extremal points of these functionals correspond to surfaces admitting minimal isometric immersions into spheres. Recently, critical metrics for the first eigenvalue were classified on tori and on Klein bottles. The present paper is concerned with extremal metrics for higher eigenvalues on these surfaces. We apply a classical construction due to Lawson. For the bipolar surface τ ˜ r , k of the Lawson's torus or Klein bottle τ r , k it is shown that: (1) If r k ≡ 0 mod 2 , τ ˜ r , k is a torus with an extremal metric for λ 4 r − 2 and λ 4 r + 2 . (2) If r k ≡ 1 mod 4 , τ ˜ r , k is a torus with an extremal metric for λ 2 r − 2 and λ 2 r + 2 . (3) If r k ≡ 3 mod 4 , τ ˜ r , k is a Klein bottle with an extremal metric for λ r − 2 and λ r + 2 . Furthermore, we find explicitly the S 1 -equivariant minimal immersion of the bipolar surfaces into S 4 by the corresponding eigenfunctions.

A new variational characterization of the Wulff shape

Abstract: Given a positive function F on S n which satisfies a convexity condition, we define the rth anisotropic mean curvature function M r for hypersurfaces in R n + 1 which is a generalization of the usual rth mean curvature function. Let X : M → R n + 1 be an n-dimensional closed hypersurface with M r + 1 M r = constant , for some r with 1 ⩽ r ⩽ n − 1 , which is a critical point for a variational problem. We show that X ( M ) is stable if and only if X ( M ) is the Wulff shape.

Singular cotangent bundle reduction & spin Calogero–Moser systems☆

Abstract: We develop a bundle picture for singular symplectic quotients of cotangent bundles acted upon by cotangent lifted actions for the case that the configuration manifold is of single orbit type. Furthermore, we give a formula for the reduced symplectic form in this setting. As an application of this bundle picture we consider Calogero–Moser systems with spin associated to polar representations of compact Lie groups.

Pseudo-Riemannian 3-manifolds with prescribed distinct constant Ricci eigenvalues

Abstract: We study three-dimensional pseudo-Riemannian manifolds having distinct constant principal Ricci curvatures. These spaces are described via a system of differential equations, and a simple characterization is proved to hold for the locally homogeneous ones. We then generalize the technique used in [O. Kowalski, F. Prufer, On Riemannian 3-manifolds with distinct constant Ricci eigenvalues, Math. Ann. 300 (1994) 17–28] for Riemannian manifolds and construct explicitly homogeneous and non-homogeneous pseudo-Riemannian metrics in R 3 , having the prescribed principal Ricci curvatures.

Relation between metric spaces and Finsler spaces

Abstract: In a connected Finsler space F n = ( M , F ) every ordered pair of points p , q ∈ M determines a distance ϱ F ( p , q ) as the infimum of the arc length of curves joining p to q . ( M , ϱ F ) is a metric space if F n is absolutely homogeneous, and it is quasi-metric space (i.e. the symmetry: ϱ F ( p , q ) = ϱ F ( q , p ) fails) if F n is positively homogeneous only. It is known the Busemann–Mayer relation lim t → t 0 + d d t ϱ F ( p 0 , p ( t ) ) = F ( p 0 , p ˙ 0 ) , for any differentiable curve p ( t ) in an F n . This establishes a 1 : 1 relation between Finsler spaces F n = ( M , F ) and (quasi-) metric spaces ( M , ϱ F ) . We show that a distance function ϱ ( p , q ) (with the differentiability property of ϱ F ) needs not to be a ϱ F . This means that the family { ( M , ϱ ) } is wider than { ( M , ϱ F ) } . We give a necessary and sufficient condition in two versions for a ϱ to be a ϱ F , i.e. for a (quasi-) metric space ( M , ϱ ) to be equivalent (with respect to the distance) to a Finsler space ( M , F ) .

Classification of complete Finsler manifolds through a second order differential equation

Abstract: By using a certain second order differential equation, the notion of adapted coordinates on Finsler manifolds is defined and some classifications of complete Finsler manifolds are found. Some examples of Finsler metrics, with positive constant sectional curvature, not necessarily of Randers type nor projectively flat, are found. This work generalizes some results in Riemannian geometry and open up, a vast area of research on Finsler geometry.

Fractional monodromy: parallel transport of homology cycles

Abstract: A 2n-dimensional completely integrable system gives rise to a singular fibration whose generic fiber is the n-torus T n . In the classical setting, it is possible to define a parallel transport of elements of the fundamental group of a fiber along a path, when the path describes a loop around a singular fiber, it defines an automorphism of π 1 ( T n ) called monodromy transformation [J.J. Duistermaat, On global action-angle coordinates, Communications on Pure and Applied Mathematics 33 (6) (1980) 687–706]. Some systems give rise to a non-classical setting, in which the path can wind around a singular fiber only by crossing a codimension 1 submanifold of special singular fibers (a wall), in this case a non-classical parallel transport can be defined on a subgroup of the fundamental group. This gives rise to what is known as monodromy with fractional coefficients [N. Nekhoroshev, D. Sadovskii, B. Zhilinskii, Fractional monodromy of resonant classical and quantum oscillators, Comptes Rendus Mathematique 335 (11) (2002) 985–988]. In this article, we give a precise meaning to the non-classical parallel transport. In particular we show that it is a homologic process and not a homotopic one. We justify this statement by describing the type of singular fibers that generate a wall that can be crossed, by describing the parallel transport in a semi-local neighbourhood of the wall of singularities, and by producing a family of 4-dimensional examples.

The symplectic normal space of a cotangent-lifted action

Abstract: For the cotangent bundle T*Q of a smooth Riemannian manifold acted upon by the lift of a smooth and proper action by isometries of a Lie group, we characterize the symplectic normal space at any point. We show that this space splits as the direct sum of the cotangent bundle of a linear space and a symplectic linear space coming from reduction of a coadjoint orbit. This characterization of the symplectic normal space can be expressed solely in terms of the group action on the base manifold and the coadjoint representation. Some relevant particular cases are explored. (C) 2007 Elsevier B.V. All rights reserved.

Fisher information metric and Poisson kernels

Abstract: A complete Riemannian manifold X with negative curvature satisfying − b 2 ⩽ K X ⩽ − a 2 0 for some constants a , b , is naturally mapped in the space of probability measures on the ideal boundary ∂X by assigning the Poisson kernels. We show that this map is embedding and the pull-back metric of the Fisher information metric by this embedding coincides with the original metric of X up to constant provided X is a rank one symmetric space of non-compact type. Furthermore, we give a geometric meaning of the embedding.

Eigenvalue estimates for Dirac operators with parallel characteristic torsion

Abstract: Assume that the compact Riemannian spin manifold ( M n , g ) admits a G -structure with characteristic connection ∇ and parallel characteristic torsion ( ∇ T = 0 ), and consider the Dirac operator D 1 / 3 corresponding to the torsion T / 3 . This operator plays an eminent role in the investigation of such manifolds and includes as special cases Kostant's “cubic Dirac operator” and the Dolbeault operator. In this article, we describe a general method of computation for lower bounds of the eigenvalues of D 1 / 3 by a clever deformation of the spinorial connection. In order to get explicit bounds, each geometric structure needs to be investigated separately; we do this in full generality in dimension 4 and for Sasaki manifolds in dimension 5.

On cylindrically bounded H-Hypersurfaces of H n × R

Abstract: We show that H-hypersurfaces of H n × R contained in a vertical cylinder and with Ricci curvature with strong quadratic decay have mean curvature | H | > ( n − 1 ) / n

Two Gauss–Bonnet and Poincaré–Hopf theorems for orbifolds with boundary

Abstract: We generalize the Gauss–Bonnet and Poincare–Hopf theorems to the case of orbifolds with boundary. We present two such generalizations, the first in the spirit of Satake, in which the local data (i.e. integral of the curvature in the case of the Gauss–Bonnet theorem and the index of the vector field in the case of the Poincare–Hopf theorem) is related to Satake's orbifold Euler–Satake characteristic, a rational number which depends on the orbifold structure. For the second pair of generalizations, we use the Chen–Ruan orbifold cohomology to express the local data in a way which can be related to the Euler characteristic of the underlying space of the orbifold.

On a class of twistorial maps

Abstract: We show that a natural class of twistorial maps gives a pattern for apparently different geometric maps, such as, ( 1 , 1 ) -geodesic immersions from ( 1 , 2 ) -symplectic almost Hermitian manifolds and pseudo horizontally conformal submersions with totally geodesic fibres for which the associated almost CR-structure is integrable. Along the way, we construct for each constant curvature Riemannian manifold ( M , g ) , of dimension m, a family of twistor spaces { Z r ( M ) } 1 ⩽ r 1 2 m such that Z r ( M ) parametrizes naturally the set of pairs ( P , J ) , where P is a totally geodesic submanifold of ( M , g ) , of codimension 2r, and J is an orthogonal complex structure on the normal bundle of P which is parallel with respect to the normal connection.

An example of coisotropic submanifolds C1-close to a given coisotropic submanifold

Abstract: We discuss a simple example of coisotropic submanifold M of a symplectic manifold, and show that the set of coisotropic submanifolds which are C1-close to M does not have a manifold structure.

Complete hypersurfaces in a Euclidean space Rn+1 with constant mth mean curvature

Abstract: In this paper, we consider n-dimensional oriented complete hypersurfaces with constant mth mean curvature of a Euclidean space R n + 1 . We characterize the hypersurface S k ( c ) × R n − k in a Euclidean space R n + 1 and show that generalized Yau conjecture is true for the class of oriented compact locally conformally flat hypersurfaces with positive constant mth mean curvature of a Euclidean space R n + 1 . When m = 2 , our results reduce to the results of Q.-M. Cheng [Q.M. Cheng, Complete hypersurfaces in a Euclidean space R n + 1 with constant scalar curvature, Indiana Univ. Math. J. 51 (2002) 53–68].

Classification of 1st order symplectic spinor operators over contact projective geometries

Abstract: We give a classification of 1st order invariant differential operators acting between sections of certain bundles associated to Cartan geometries of the so-called metaplectic contact projective type. These bundles are associated via representations, which are derived from the so-called higher symplectic (sometimes also called harmonic or generalized Kostant) spinor modules. Higher symplectic spinor modules are arising from the Segal–Shale–Weil representation of the metaplectic group by tensoring it by finite dimensional modules. We show that for all pairs of the considered bundles, there is at most one 1st order invariant differential operator up to a complex multiple and give an equivalence condition for the existence of such an operator. Contact projective analogues of the well known Dirac, twistor and Rarita–Schwinger operators appearing in Riemannian geometry are special examples of these operators.

Higher order parallel surfaces in the Heisenberg group

Abstract: We give a classification of k-parallel surfaces in the three-dimensional Heisenberg group. In particular, we prove that every k-parallel surface in the Heisenberg group is a vertical cylinder over a polynomial spiral of degree at most k − 1 .

Lie solutions of Riemannian transport equations on compact manifolds

Abstract: Given a couple of smooth positive measures of same total mass on a compact Riemannian manifold, the associated optimal transport equation admits a symplectic Monge–Ampere structure, hence Lie solutions (in a restricted sense, though, still expressing measure-transport). Properties of such solutions are recorded; a structure result is obtained for regular ones (each consisting of a closed 1-form composed with a diffeomorphism) and a quadratic cost-functional proposed for them.