On focal locus of submanifolds of naturally reductive compact Riemannian homogeneous spaces

The purpose of this paper is to discuss focal points of submanifolds of naturally reductive compact Riemannian homogeneous spaces. Focal points have been studied by other workers [1, 4, 7]. In this paper we have estimated location of focal points along geodesic and obtained a condition under which a point is a focal point along geodesic.     

Einstein metrics on compact Lie groups which are not naturally reductive

The study of left-invariant Einstein metrics on compact Lie groups which are naturally reductive was initiated by D??Atri and Ziller (Mem Am Math Soc 18, (215) 1979). In 1996 the second author obtained non-naturally reductive Einstein metrics on the Lie group SU(n) for n ??? 6, by using a method of Riemannian submersions. In the present work we prove existence of non-naturally reductive Einstein metrics on the compact simple Lie groups SO(n) (n ??? 11), Sp(n) (n ??? 3), E ₆, E ₇, and E ₈.     

Canonical connections with an algebraic curvature tensor field on naturally reductive spaces

We study Ambrose-Singer connections with an algebraic curvature tensor on simply connected manifolds carrying a homogeneous Riemannian structure of class ₃ in the classification given by F. Tricerri and L. Vanhecke.This work was partially supported by MURST.     

The automorphism groups of compact homogeneous spaces

Considering the automorphism groups of compact homogeneous spaces we inspect certain general properties, indicate a method for calculating the groups, and illustrate it with examples in a few particular cases.     

The comb representation of compact ultrametric spaces

We call a comb a map f: I → [0,∞), where I is a compact interval, such that {f ≥ ε} is finite for any ε > 0. A comb induces a (pseudo)-distance \({\overline d _f}\) on {f = 0} defined by \({\overline d _f}\left( {s,t} \right) = {\max _{\left( {s \wedge t,s \vee t} \right)}}f\). We describe the completion \(\overline I \) of {f = 0} for this metric, which is a compact ultrametric space called the comb metric space.Conversely, we prove that any compact, ultrametric space (U, d) without isolated points is isometric to a comb metric space. We show various examples of the comb representation of well-known ultrametric spaces: the Kingman coalescent, infinite sequences of a finite alphabet, the p-adic field and spheres of locally compact real trees. In particular, for a rooted, locally compact real tree defined from its contour process h, the comb isometric to the sphere of radius T centered at the root can be extracted from h as the depths of its excursions away from T.     

Indestructibility of compact spaces

In this article we investigate which compact spaces remain compact under countably closed forcing. We prove that, assuming the Continuum Hypothesis, the natural generalizations to _ω1${\omega }_1$-sequences of the selection principle and topological game versions of the Rothberger property are not equivalent, even for compact spaces. We also show that Tall and Usuba?s “_ℵ1${\aleph }_1$-Borel Conjecture” is equiconsistent with the existence of an inaccessible cardinal.     

Naturally reductive homogeneous Finsler spaces

In this paper, we study naturally reductive Finsler metrics. We first give a sufficient and necessary condition for a Finsler metric to be naturally reductive with respect to certain transitive group of isometries. Then we study in detail the left invariant naturally reductive metrics on compact Lie groups and give a method to construct the non-Riemannian ones. Further, we give a classification of left invariant naturally reductive metrics on nilpotent Lie groups. Finally, we give a classification of all the naturally reductive Finsler spaces of dimension less or qual to 4. As applications, we obtain some rigidity theorems about naturally reductive Finsler metrics. Namely, any left invariant non-symmetric naturally reductive Finsler metric on a compact simple Lie group or an indecomposable nilpotent Lie group must be Riemannian. On the other hand, we provide a very convenient method to construct non-symmetric Berwald spaces which are neither Riemannian nor locally Minkowskian, a kind of spaces which are sought after in the book by Bao et al. (An introduction to Riemann–Finsler geometry, GTM 200, 2000).     

The Lichnerowicz Laplacian on compact symmetric spaces

We show that, on a compact symmetric space, the Lichnerowicz Laplacian acting on the space of covariant tensor fields coincides with the Casimir operator and we deduce that, on a compact semisimple Lie group, the Lichnerowicz Laplacian is the mean of the left invariant Casimir operator and the right invariant Casimir operator. To cite this article: M. Boucetta, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Small Valdivia compact spaces

We prove a preservation theorem for the class of Valdivia compact spaces, which involves inverse sequences of retractions of a certain kind. Consequently, a compact space of weight?ℵ₁ is Valdivia compact iff it is the limit of an inverse sequence of metric compacta whose bonding maps are retractions. As a corollary, we show that the class of Valdivia compacta of weight?ℵ₁ is preserved both under retractions and under open 0-dimensional images. Finally, we characterize the class of all Valdivia compacta in the language of category theory, which implies that this class is preserved under all continuous weight preserving functors.     

The moduli space of hyperbolic compact complex spaces

In this paper, we construct the moduli space of reduced hyperbolic compact complex spaces. First, we prove an infinitesimal characterization of hyperbolicity using a family of Kobayashi–Royden pseudo-metrics introduced by Venturini and as a consequence we conclude that the property of Landau holds for complex spaces. Finally, we establish this moduli space in the case of locally trivial deformations, and in a more general situation, the case of equisingular deformations.     

Isometric embeddings of compact spaces into Banach spaces

We show the existence of a compact metric space K such that whenever K embeds isometrically into a Banach space Y, then any separable Banach space is linearly isometric to a subspace of Y. We also address the following related question: if a Banach space Y contains an isometric copy of the unit ball or of some special compact subset of a separable Banach space X, does it necessarily contain a subspace isometric to X? We answer positively this question when X is a polyhedral finite-dimensional space, c₀ or ?₁.     

Torus symmetry of compact self-dual manifolds

We classify compact anti-self-dual Hermitian surfaces and compact four-dimensional conformally flat manifolds for which the group of orientation preserving conformal transformations contains a two-dimensional torus. As a corollary, we derive a topological classification of compact self-dual manifolds for which the group of conformal transformations contains a two-dimensional torus.Partially supported by the National Science Foundation grant DMS-9306950.