Some Remarks Concerning Hyperholomorphic B-Manifolds

The authors consider a differentiable manifold with H-structure which is an isomorphic representation of an associative, commutative and unitial algebra. For Riemannian metric tensor fields, the φ-operators associated with r-regular H-structure are introduced. With the help of φ-operators, the hyperholomorphity condition of B-manifolds is established.     

The metric geometry of the manifold of Riemannian metrics over a closed manifold

We prove that the L ² Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold induces a metric space structure. As the L ² metric is a weak Riemannian metric, this fact does not follow from general results. In addition, we prove several results on the exponential mapping and distance function of a weak Riemannian metric on a Hilbert/Fréchet manifold. The statements are analogous to, but weaker than, what is known in the case of a Riemannian metric on a finite-dimensional manifold or a strong Riemannian metric on a Hilbert manifold.     

On the topology and the geometry of SO(3)-manifolds

Consider the non-standard embedding of SO(3) into SO(5) given by the five-dimensional irreducible representation of SO(3), henceforth called SO(3)_ir. In this note, we study the topology and the differential geometry of five-dimensional Riemannian manifolds carrying such an SO(3)_ir structure, i.e., with a reduction of the frame bundle to SO(3)_ir.     

Totally geodesic maps into metric spaces

We prove that a totally geodesic map between a Riemannian manifold and a metric space can be represented as the composite of a totally geodesic map from a Riemannian manifold to a Finslerian manifold and a locally isometric embedding between metric spaces. As a corollary, we obtain the homotheticity of a totally geodesic map from an irreducible Riemannian manifold to an Alexandrov space of curvature bounded above. This is a generalization of the case between Riemannian manifolds. Mathematics Subject Classification (2000): 53C20, 53C22, 53C24 Received: 14 March 2002; in final form: 6 May 2002 / / Published online: 24 February 2003     

Reconstructing the geometric structure of a Riemannian symmetric space from its Satake diagram

The local geometry of a Riemannian symmetric space is described completely by the Riemannian metric and the Riemannian curvature tensor of the space. In the present article I describe how to compute these tensors for any Riemannian symmetric space from its Satake diagram, in a way that is suited for the use with computer algebra systems; an example implementation for Maple Version 10 can be found on . As an example application, the totally geodesic submanifolds of the Riemannian symmetric space SU(3)/SO(3) are classified.      

Killing tensor fields on the 2-torus

A symmetric tensor field on a Riemannian manifold is called a Killing field if the symmetric part of its covariant derivative equals zero. There is a one-to-one correspondence between Killing tensor fields and first integrals of the geodesic flow which depend polynomially on the velocity. Therefore Killing tensor fields relate closely to the problem of integrability of geodesic flows. In particular, the following question is still open: does there exist a Riemannian metric on the 2-torus which admits an irreducible Killing tensor field of rank ≥ 3? We obtain two necessary conditions on a Riemannian metric on the 2-torus for the existence of Killing tensor fields. The first condition is valid for Killing tensor fields of arbitrary rank and relates to closed geodesics. The second condition is obtained for rank 3 Killing tensor fields and pertains to isolines of the Gaussian curvature.     

A vanishing result for the Spin c Dirac operator defined along the leaves of a foliation

In the setting of a closed Riemannian manifold endowed with a smooth, non-necessarily integrable distribution, we extend a Lichnerowicz type formula which is known to work in the particular case of a transverse bundle associated to a Riemannian foliation. Interesting settings in which non-integrable distributions appear naturally are emphasized. As an application, we consider the distribution as being even dimensional and integrable; we consider also a hermitian line bundle, with a hermitian connection, such that the induced curvature tensor is non-degenerate, and an arbitrary hermitian bundle endowed also with a hermitian connection. Taking the k power of the line bundle and canonically constructing a Spin ^c Dirac operator defined along the leaves of the foliation generated by the distribution, we prove a vanishing result for the half kernel of this operator.     

Normality of the Twistor Space of a 5-Manifold with an Irreducible SO(3)-Structure

Mediterranean Journal of Mathematics - A manifold with an irreducible SO(3)-structure is a 5-manifold M whose structure group can be reduced to the group SO(3), non-standardly imbedded in SO(5)....     

Geometric properties of almost contact metric 3-submersions

Summary In this paper, we discuss some geometric properties of three types of Riemannian submersions whose total space is an almost contact metric manifold with 3-structure. The study is focused on the transference of structures.     

Holomorphically projective transformations on complex Riemannian manifold

The aim of the paper is to prove that if a complex Riemannian manifold with holomorphic characteristic connection is holomorphically projective equivalent to a locally symmetric space then it is a complex Riemannian manifold of pointwise constant holomorphic characteristic sectional curvature.Dedicated to N.K. Stephanidis on the occasion of his 65 th birthday.     

Integrable Riemannian Submersion with Singularities

A map of a Riemannian manifold into an euclidian space is said to be transnormal if its restrictions to neighbourhoods of regular level sets are integrable Riemannian submersions. Analytic transnormal maps can be used to describe isoparametric submanifolds in spaces of constant curvature and equifocal submanifolds with flat sections in simply connected symmetric spaces. These submanifolds are also regular leaves of singular Riemannian foliations with sections. We prove that regular level sets of an analytic transnormal map on a real analytic complete Riemannian manifold are equifocal submanifolds and leaves of a singular Riemannian foliation with sections.     

Travel Time Tomography

We survey some results on travel time tomography. The question is whether we can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as geometry problems, the boundary rigidity problem and the lens rigidity problem. The boundary rigidity problem is whether we can determine a Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function between boundary points. The lens rigidity problem problem is to determine a Riemannian metric of a Riemannian manifold with boundary by measuring for every point and direction of entrance of a geodesic the point of exit and direction of exit and its length. The linearization of these two problems is tensor tomography. The question is whether one can determine a symmetric two-tensor from its integrals along geodesics. We emphasize recent results on boundary and lens rigidity and in tensor tomography in the partial data case, with further applications.     

Ricci张量对称函数的预定问题

本文考虑Ricci张量的对称函数σ2(Ricg)的预定问题.假设(M,g)是闭的Einstein流形,我们得到了只要流形(M,g)不具有σ2(Ric)奇性,则对于变号的函数f∈C∞(M),存在度量g*,使得σ2(Ricg*) = f.然后,作为推论,得到了具有负数量曲率的闭Einstein流形上的预定曲率的结果.     

Riesz transform and perturbation

We prove that, on a complete noncompact Riemannian manifold with bounded geometry, the L^p boundedness of the Riesz transform, for p>2, is stable under a quasi-isometric and integrable change of metric. As an intermediate step, we treat the case of weighted divergence form operators in the Euclidean space.     

Regularity of ghosts in tensor tomography

We study on a compact Riemannian manifold with boundary the ray transform I which integrates symmetric tensor fields over geodesics. A tensor field is said to be a nontrivial ghost if it is in the kernel of I and is L²-orthogonal to all potential fields. We prove that a nontrivial ghost is smooth in the case of a simple metric. This implies that the wave front set of the solenoidal part of a field f can be recovered from the ray transform If. We give an explicit procedure for recovering the wave front set.     

Conformally Osserman manifolds of dimension 16 and a Weyl–Schouten theorem for rank-one symmetric spaces

A Riemannian manifold is called Osserman (conformally Osserman, respectively), if the eigenvalues of the Jacobi operator of its curvature tensor (Weyl tensor, respectively) are constant on the unit tangent sphere at every point. The Osserman Conjecture asserts that any Osserman manifold is either flat or rank-one symmetric. We prove that both the Osserman Conjecture and its conformal version, the Conformal Osserman Conjecture, are true, modulo a certain assumption on algebraic curvature tensors in ${\mathbb {R}^{16}}$ . As a consequence, we show that a Riemannian manifold having the same Weyl tensor as a rank-one symmetric space is conformally equivalent to it.     

A classification of hyperpolar and cohomogeneity one actions

An isometric action of a compact Lie group on a Riemannian manifold is called hyperpolar if there exists a closed, connected submanifold that is flat in the induced metric and meets all orbits orthogonally. In this article, a classification of hyperpolar actions on the irreducible Riemannian symmetric spaces of compact type is given. Since on these symmetric spaces actions of cohomogeneity one are hyperpolar, i.e. normal geodesics are closed, we obtain a classification of the homogeneous hypersurfaces in these spaces by computing the cohomogeneity for all hyperpolar actions. This result implies a classification of the cohomogeneity one actions on compact strongly isotropy irreducible homogeneous spaces.

     

Nonholonomic Riemann and Weyl tensors for flag manifolds

On any manifold, any nondegenerate symmetric 2-form (metric) and any nondegenerate skew-symmetric differential form ω can be reduced to a canonical form at any point but not in any neighborhood: the corresponding obstructions are the Riemannian tensor and dω. The obstructions to flatness (to reducibility to a canonical form) are well known for any G-structure, not only for Riemannian or almost symplectic structures. For a manifold with a nonholonomic structure (nonintegrable distribution), the general notions of flatness and obstructions to it, although of huge interest (e.g., in supergravity) were not known until recently, although particular cases have been known for more than a century (e.g., any contact structure is nonholonomically “flat”: it can always be reduced locally to a canonical form). We give a general definition of the nonholonomic analogues of the Riemann tensor and its conformally invariant analogue, the Weyl tensor, in terms of Lie algebra cohomology and quote Premet’s theorems describing these cohomologies. Using Premet’s theorems and the SuperLie package, we calculate the tensors for flag manifolds associated with each maximal parabolic subalgebra of each simple Lie algebra (and in several more cases) and also compute the obstructions to flatness of the G(2)-structure and its nonholonomic superanalogue. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 2, pp. 186–219, November, 2007.     

Cheeger-gromoll type metrics on the (1,1)-tensor bundles

Using a Riemannian metric on a differentiable manifold, a Cheeger-Gromoll type metric is introduced on the (1,1)-tensor bundle of the manifold. Then the Levi-Civita connection, Riemannian curvature tensor, Ricci tensor, scalar curvature and sectional curvature of this metric are calculated. Also, a para-Nordenian structure on the the (1,1)-tensor bundle with this metric is constructed and the geometric properties of this structure are studied.     

The Ricci flow as a geodesic on the manifold of Riemannian metrics

The Ricci flow is an evolution equation in the space of Riemannian metrics.A solution for this equation is a curve on the manifold of Riemannian metrics. In this paper we introduce a metric on the manifold of Riemannian metrics such that the Ricci flow becomes a geodesic.We show that the Ricci solitons introduce a special slice on the manifold of Riemannian metrics.