On δ-homogeneous Riemannian manifolds

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.     

On <Emphasis Type="Italic">δ</Emphasis>-homogeneous Riemannian manifolds. II

We continue the study of the δ-homogeneous Riemannian manifolds defined in a more general case by V. N. Berestovski? and C. P. Plaut. Each of these manifolds has nonnegative sectional curvature. We prove in particular that every naturally reductive compact homogeneous Riemannian manifold of positive Euler characteristic is δ-homogeneous.     

Three- and Four-Dimensional Einstein-like Manifolds and Homogeneity

The aim of this paper is to study three- and four-dimensional Einstein-like Riemannian manifolds which are Ricci-curvature homogeneous, that is, have constant Ricci eigenvalues. In the three-dimensional case, we present the complete classification of these spaces while, in the four-dimensional case, this classification is obtained in the special case where the manifold is locally homogeneous. We also present explicit examples of four-dimensional locally homogeneous Riemannian manifolds whose Ricci tensor is cyclic-parallel (that is, are of type A) and has distinct eigenvalues. These examples are invalidating an expectation stated by F. Podestá and A. Spiro, and illustrating a striking contrast with the three-dimensional case (where this situation cannot occur). Finally, we also investigate the relation between three- and four-dimensional Einstein-like manifolds of type A and D'Atri spaces, that is, Riemannian manifolds whose geodesic symmetries are volume-preserving (up to sign).     

On geometric vector fields of Minkowski spaces and their applications

As it is well-known, a Minkowski space is a finite dimensional real vector space equipped with a Minkowski functional F. By the help of its second order partial derivatives we can introduce a Riemannian metric on the vector space and the indicatrix hypersurface S:=F⁻¹(1) can be investigated as a Riemannian submanifold in the usual sense.Our aim is to study affine vector fields on the vector space which are, at the same time, affine with respect to the Funk metric associated with the indicatrix hypersurface. We give an upper bound for the dimension of their (real) Lie algebra and it is proved that equality holds if and only if the Minkowski space is Euclidean. Criteria of the existence is also given in lower dimensional cases. Note that in case of a Euclidean vector space the Funk metric reduces to the standard Cayley-Klein metric perturbed with a nonzero 1-form.As an application of our results we present the general solution of Matsumoto's problem on conformal equivalent Berwald and locally Minkowski manifolds. The reasoning is based on the theory of harmonic vector fields on the tangent spaces as Riemannian manifolds or, in an equivalent way, as Minkowski spaces. Our main result states that the conformal equivalence between two Berwald manifolds must be trivial unless the manifolds are Riemannian.     

A Class of Homogeneous Einstein Manifolds

A Riemannian manifold (M, g) is called Einstein manifold if its Ricci tensor satisfies r = c·g for some constant c. General existence results are hard to obtain, e.g., it is as yet unknown whether every compact manifold admits an Einstein metric. A natural approach is to impose additional homogeneous assumptions. M. Y. Wang and W. Ziller have got some results on compact homogeneous space G/H. They investigate standard homogeneous metrics, the metric induced by Killing form on G/H, and get some classification results. In this paper some more general homogeneous metrics on some homogeneous space G/H are studies, and a necessary and sufficient condition for this metric to be Einstein is given. The authors also give some examples of Einstein manifolds with non-standard homogeneous metrics.     

带位势的p-调和热流

在本文中我们考察从黎曼流形M到具左不变度量的齐次空间N的带位势的 p-调和映射热流,证明存在全局弱解.     

Compactness for manifolds and integral currents with bounded diameter and volume

By Gromov??s compactness theorem for metric spaces, every uniformly compact sequence of metric spaces admits an isometric embedding into a common compact metric space in which a subsequence converges with respect to the Hausdorff distance. Working in the class of oriented k-dimensional Riemannian manifolds (with boundary) and, more generally, integral currents in metric spaces in the sense of Ambrosio?CKirchheim and replacing the Hausdorff distance with the filling volume or flat distance, we prove an analogous compactness theorem in which however we only assume uniform bounds on volume and diameter.     

Homogeneous spaces of curvature bounded below

We prove that every locally connected quotient G/H of a locally compact, connected, first countable topological group G by a compact subgroup H admits a G-invariant inner metric with curvature bounded below. Every locally compact homogeneous space of curvature bounded below is isometric to such a space. These metric spaces generalize the notion of Riemannian homogeneous space to infinite dimensional groups and quotients which are never (even infinite dimensional) manifolds. We study the geometry of these spaces, in particular of non-negatively curved homogeneous spaces. Dedicated to the memory of A. D. Alexandrov     

KILLING VECTOR FIELDS OF CONSTANT LENGTH ON LOCALLY SYMMETRIC RIEMANNIAN MANIFOLDS

In this paper nontrivial Killing vector fields of constant length and the corresponding ows on smooth complete Riemannian manifolds are investigated. It is proved that such a ow on symmetric space is free or induced by a free isometric action of the circle S ¹. Examples of unit Killing vector fields generated by almost free but not free actions of S ¹ on locally symmetric Riemannian spaces are found; among them are homogeneous (nonsimply connected) Riemannian manifolds of constant positive sectional curvature and locally Euclidean spaces. Some unsolved questions are formulated. DOI: .     

Nonlinear Bound States on Weakly Homogeneous Spaces

We prove the existence of ground state solutions for a class of nonlinear elliptic equations, arising in the production of standing wave solutions to an associated family of nonlinear Schrödinger equations. We examine two constrained minimization problems, which give rise to such solutions. One yields what we call F _λ-minimizers, the other energy minimizers. We produce such ground state solutions on a class of Riemannian manifolds called weakly homogeneous spaces, and establish smoothness, positivity, and decay properties. We also identify classes of Riemannian manifolds with no such minimizers, and classes for which essential uniqueness of positive solutions to the associated elliptic PDE fails.     

Metric projections versus non-positive curvature

In this paper two metric properties on geodesic length spaces are introduced by means of the metric projection, studying their validity on Alexandrov and Busemann NPC spaces. In particular, we prove that both properties characterize the non-positivity of the sectional curvature on Riemannian manifolds. Further results are also established on reversible/non-reversible Finsler–Minkowski spaces.     

Quantitative property A, Poincaré inequalities, Lp-compression and Lp-distortion for metric measure spaces

We introduce a quantitative version of Property A in order to estimate the L ^p -compressions of a metric measure space X. We obtain various estimates for spaces with sub-exponential volume growth. This quantitative property A also appears to be useful to yield upper bounds on the L ^p -distortion of finite metric spaces. Namely, we obtain new optimal results for finite subsets of homogeneous Riemannian manifolds. We also introduce a general form of Poincaré inequalities that provide constraints on compressions, and lower bounds on distortion. These inequalities are used to prove the optimality of some of our results.      

Invariant <Emphasis Type="Italic">f</Emphasis>-structures on naturally reductive homogeneous spaces

We study invariant metric f-structures on naturally reductive homogeneous spaces and establish their relation to generalized Hermitian geometry. We prove a series of criteria characterizing geometric and algebraic properties of important classes of metric f-structures: nearly Kähler, Hermitian, Kähler, and Killing structures. It is shown that canonical f-structures on homogeneous Φ-spaces of order k (homogeneous k-symmetric spaces) play remarkable part in this line of investigation. In particular, we present the final results concerning canonical f-structures on naturally reductive homogeneous Φ-spaces of order 4 and 5.     

Minimally separating sets, mediatrices, and Brillouin spaces

Brillouin zones and their boundaries were studied in [J.J.P. Veerman et al., Comm. Math. Phys. 212 (3) (2000) 725] because they play an important role in focal decomposition as first defined by Peixoto in [J. Differential Equations 44 (1982) 271] and in physics [N.W. Ashcroft, N.D. Mermin, Solid State Physics, Holt, Rhinehart, and Winston, 1976; L. Brillouin, Wave Propagation in Periodic Structures, Dover, 1953]. In so-called Brillouin spaces, the boundaries of the Brillouin zones have certain regularity properties which imply that they consist of pieces of mediatrices (or equidistant sets).The purpose of this note is two-fold. First, we give some simple conditions on a metric space which are sufficient for it to be a Brillouin space. These conditions show, for example, that all compact, connected Riemannian manifolds with their usual distance functions are Brillouin spaces. Second, we exhibit a restriction on the Z₂-homology of mediatrices in such manifolds in terms of the Z₂-homology of the manifolds themselves, based on the fact that they are Brillouin spaces. (This will used to obtain a classification up to homeomorphism of surface mediatrices in forthcoming paper [J. Bernhard, J.J.P. Veerman, The topology of surface mediatrices, Portland State University].)     

Stability problems in a theorem of F. Schur

Schur's theorem states that an isotropic Riemannian manifold of dimension greater than two has constant curvature. It is natural to guess that compact almost isotropic Riemannian manifolds of dimension greater than two are close to spaces of almost constant curvature. We take the curvature anisotropy as the discrepancy of the sectional curvatures at a point. The main result of this paper is that Riemannian manifolds in Cheeger's class ℜ(n,d,V,A) withL ₁-small integral anisotropy haveL _p-small change of the sectional curvature over the manifold. We also estimate the deviation of the metric tensor from that of constant curvature in theW _p ² -norm, and prove that compact almost isotropic spaces inherit the differential structure of a space form. These stability results are based on the generalization of Schur' theorem to metric spaces.     

Invariant affinor and sub-Kähler structures on homogeneous spaces

We consider G-invariant affinor metric structures and their particular cases, sub-Kähler structures, on a homogeneous space G/H. The affinor metric structures generalize almost Kähler and almost contact metric structures to manifolds of arbitrary dimension. We consider invariant sub-Riemannian and sub-Kähler structures related to a fixed 1-form with a nontrivial radical. In addition to giving some results for homogeneous spaces of arbitrary dimension, we study these structures separately on the homogeneous spaces of dimension 4 and 5.     

Discrete Approximation of Spaces of Homogeneous Type

In this note we combine the dyadic families introduced by M. Christ in (Colloq. Math. 60/61(2):601–628, 1990) and the discrete partitions introduced by J.M. Wu in (Proc. Am. Math. Soc. 126(5):1453–1459, 1998) to get approximation of a compact space of homogeneous type by a uniform sequence of finite spaces of homogeneous type. The convergence holds in the sense of a metric built on the Hausdorff distance between compact sets and on the Kantorovich-Rubinshtein metric between measures. The authors were supported by CONICET, CAI+D (UNL) and ANPCyT.     

Conjugate points in length spaces

In this paper we extend the concept of a conjugate point in a Riemannian manifold to geodesic spaces. In particular, we introduce symmetric conjugate points and ultimate conjugate points and relate these notions to prior notions developed for more restricted classes of spaces. We generalize the long homotopy lemma of Klingenberg to this setting as well as the injectivity radius estimate also due to Klingenberg which was used to produce closed geodesics or conjugate points on Riemannian manifolds. We close with applications of these new kinds of conjugate points to CBA(κ) spaces: proving both known and new theorems. In particular we prove a Rauch comparison theorem, a Relative Rauch Comparison Theorem, the fact that there are no ultimate conjugate points less than π apart in a CBA(1) space and a few facts concerning closed geodesics. This paper is written to be accessible to students and includes open problems.     

Locally conformally homogeneous pseudo-Riemannian spaces

Locally homogeneous Riemannian spaces were studied in [1–4]. Locally conformally homogeneous Riemannian spaces were considered in [10]. Moreover, the theorem claiming that every such space is either conformally flat or conformally equivalent to a locally homogeneous Riemannian space was proved.In this article, we study locally conformally homogeneous pseudo-Riemannian spaces and prove a theorem on their structure. Using three-dimensional Lie groups and the six-dimensional Heisenberg group [11], we construct some examples showing the difference between the Riemannian and pseudo-Riemannian cases for such spaces.