Sobolev spaces with variable exponents on Riemannian manifolds

In this article we study variable exponent Sobolev spaces on Riemannian manifolds. The spaces are examined in the case of compact manifolds. Continuous and compact embeddings are discussed. The paper contains an example of the application of the theory to elliptic equations on compact manifolds.     

Homogeneous nearly K?hler manifolds

The structure of nearly K?hler manifolds was studied by Gray in several articles, mainly in Gray (Math Ann 223:233?C248, 1976). More recently, a relevant progress on the subject has been done by Nagy. Among other results, he proved that a complete strict nearly K?hler manifold is locally a Riemannian product of homogeneous nearly K?hler spaces, twistor spaces over quaternionic K?hler manifolds and six-dimensional (6D) nearly K?hler manifolds, where the homogeneous nearly K?hler factors are also 3-symmetric spaces. In the present article, we show some further properties relative to the structure of nearly K?hler manifolds and, using the lists of 3-symmetric spaces given by Wolf and Gray, we display the exhaustive list of irreducible simply connected homogeneous strict nearly K?hler manifolds. For such manifolds, we give details relative to the intrinsic torsion and the Riemannian curvature.     

Homogeneity and Curvatures of Geodesic Spheres

We study some scalar curvature invariants on geodesic spheres and use them to characterize several kinds of Riemannian manifolds such as homogenous manifolds and in particular, the two-point homogeneous spaces and the Damek-Ricci spaces.     

Generalised connected sums of quaternionic manifolds

Using deformations of singular twistor spaces, a generalisation of the connected sum construction appropriate for quaternionic manifolds is introduced. This is used to construct examples of quaternionic manifolds which have no quaternionic symmetries and leads to examples of quaternionic manifolds whose twistor spaces have arbitrary algebraic dimension.Partially supported by the National Science Foundation grant DMS-9296168.     

U(1)-decomposable self-dual manifolds

We characterize conformally flat spaces as the only compact self-dual manifolds which are U(1)-equivariantly and conformally decomposable into two complete self-dual Einstein manifolds with common conformal infinity. A geometric characterization of such conformally flat spaces is also given.     

Isotropy Subgroups of Transformation Groups

In this paper we consider transitive actions of Lie groups on analytic manifolds. We study three cases of analytic manifolds and their corresponding transformation groups. Given a free action on the left, we define left orbit spaces and consider actions on the right by maximal compact subgroups. We show that these actions are transitive and find the corresponding isotropy subgroups. Further, we show that the left orbit spaces are reductive homogeneous spaces. This article thus forms the basis of a forthcoming paper on invariant differential operators on homogeneous manifolds. Partially supported by a Carver Research Initiative Grant.     

A quasifibration of spaces of positive scalar curvature metrics

In this paper we show that for Riemannian manifolds with boundary the natural restriction map is a quasifibration between spaces of metrics of positive scalar curvature. We apply this result to study homotopy properties of spaces of such metrics on manifolds with boundary.

     

几何分析中运用Gromov方法的几个课题

近３０年来Ｇｒｏｍｏｖ对数学的多个领域，其中包括微分几何，拓扑，动力系统，群论和偏微分方程，作出了重要的贡献，本文讨论几何分析中与Ｇｒｏｍｏｖ引进的多种几何不变量有关的几个专题中主要包括Ｇｒｏｍｏｖ几乎平坦流形，极小体积空隙独测，填充黎曼流形，等周不等式，Ｇｒｏｍｏｖ字双曲群，非紧空间和具有界曲率奇异空间上的加权Ｌ＾ｐ上同调。     

Twistor theory for co-CR quaternionic manifolds and related structures

In a general and non-metrical framework, we introduce the class of co-CR quaternionic manifolds, which contains the class of quaternionic manifolds, whilst in dimension three it particularizes to give the Einstein-Weyl spaces. We show that these manifolds have a rich natural Twistor Theory and, along the way, we obtain a heaven space construction for quaternionic-Kähler manifolds.     

Asymptotic Dirichlet Problem for Harmonic Maps with Bounded Image

Existence and uniqueness is proved for asymptotic Dirichlet problems on Hadamard manifolds. This includes manifolds of bounded negative curvature and symmetric spaces of higher rank.     

Concircular vector fields on semi-Riemannian spaces

In this paper, we construct an analogue of concircular fields for semi-Riemannian spaces (i.e., for manifolds with degenerate metrics). We find a tensor criterion of spaces admitting the maximal number of concircular fields or having no such fields. We detect a gap in the distribution of dimensions of the space of concircular fields, which, in contrast to the corresponding gap in the case of pseudo-Riemannian manifolds, is lesser by 1. We also study some special types of concircular fields having no analogues for pseudo-Riemannian manifolds. The canonical form of the metric for some classes of semi-Riemannian spaces admitting concircular fields is obtained. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 31, Geometry, 2005.     

Global Weak Rigidity of the Gauss–Codazzi–Ricci Equations and Isometric Immersions of Riemannian Manifolds with Lower Regularity

We are concerned with the global weak rigidity of the Gauss–Codazzi–Ricci (GCR) equations on Riemannian manifolds and the corresponding isometric immersions of Riemannian manifolds into the Euclidean spaces. We develop a unified intrinsic approach to establish the global weak rigidity of both the GCR equations and isometric immersions of the Riemannian manifolds, independent of the local coordinates, and provide further insights of the previous local results and arguments. The critical case has also been analyzed. To achieve this, we first reformulate the GCR equations with div-curl structure intrinsically on Riemannian manifolds and develop a global, intrinsic version of the div-curl lemma and other nonlinear techniques to tackle the global weak rigidity on manifolds. In particular, a general functional-analytic compensated compactness theorem on Banach spaces has been established, which includes the intrinsic div-curl lemma on Riemannian manifolds as a special case. The equivalence of global isometric immersions, the Cartan formalism, and the GCR equations on the Riemannian manifolds with lower regularity is established. We also prove a new weak rigidity result along the way, pertaining to the Cartan formalism, for Riemannian manifolds with lower regularity, and extend the weak rigidity results for Riemannian manifolds with corresponding different metrics.     

KILLING VECTOR FIELDS OF CONSTANT LENGTH ON RIEMANNIAN NORMAL HOMOGENEOUS SPACES

Killing vector fields of constant length correspond to isometries of constant displacement. Those in turn have been used to study homogeneity of Riemannian and Finsler quotient manifolds. Almost all of that work has been done for group manifolds or, more generally, for symmetric spaces. This paper extends the scope of research on constant length Killing vector fields to a class of Riemannian normal homogeneous spaces.     

Sobolev spaces on Riemannian manifolds with bounded geometry: General coordinates and traces

We study fractional Sobolev and Besov spaces on noncompact Riemannian manifolds with bounded geometry. Usually, these spaces are defined via geodesic normal coordinates which, depending on the problem at hand, may often not be the best choice. We consider a more general definition subject to different local coordinates and give sufficient conditions on the corresponding coordinates resulting in equivalent norms. Our main application is the computation of traces on submanifolds with the help of Fermi coordinates. Our results also hold for corresponding spaces defined on vector bundles of bounded geometry and, moreover, can be generalized to Triebel‐Lizorkin spaces on manifolds, improving [11].     

A Class of Metric Spaces Which Do Not Coarsely Contain Expanders

In this paper, a class of metric spaces which include Hilbert spaces and Hadamard manifolds are defined. And the expanders cannot be coarsely embedded into this class of metric spaces are proved.     

Sobolev spaces and harmonic maps between singular spaces

Recently Korevaar and Schoen developed a Sobolev theory for maps from smooth (at least ) manifolds into general metric spaces by proving that the weak limit of appropriate average difference quotients is well behaved. Here we extend this theory to functions defined over Lipschitz manifold. As an application we then prove an existence theorem for harmonic maps from Lipschitz manifolds to NPC metric spaces. Received December 6, 1996 / Accepted March 4, 1997     

Detecting Hilbert manifolds among isometrically homogeneous metric spaces

We detect Hilbert manifolds among isometrically homogeneous metric spaces and apply the obtained results to recognizing Hilbert manifolds among homogeneous spaces of the form G/H, where G is a metrizable topological group and H is a closed balanced subgroup of G.     

Index of Sobolev problems on manifolds with many-dimensional singularities

We consider Sobolev spaces on manifolds with many-dimensional singularities. We prove the Fredholm property of such problems and derive the corresponding index formula. The results are based on the theory of translators on manifolds with singularities.