Compact Homogeneous Riemannian Manifolds Research Articles

On anti-quasi-Sasakian manifolds of maximal rank

We discuss the existence of invariant anti-quasi-Sasakian (aqS) structures of maximal rank on compact homogeneous Riemannian manifolds and on nilpotent Lie groups. In the former case we obtain a non-existence result, while in the latter case we provide a complete classification. We also show that every compact aqS manifold has nonvanishing second Betti number.

Bismut Ricci flat manifolds with symmetries

We construct examples of compact homogeneous Riemannian manifolds admitting an invariant Bismut connection that is Ricci flat and non-flat, proving in this way that the generalized Alekseevsky–Kimelfeld theorem does not hold. The classification of compact homogeneous Bismut Ricci flat spaces in dimension$5$is also provided. Moreover, we investigate compact homogeneous spaces with non-trivial third Betti number, and we point out other possible ways to construct Bismut Ricci flat manifolds. Finally, since Bismut Ricci flat connections correspond to fixed points of the generalized Ricci flow, we discuss the stability of some of our examples under the flow.

The radii of sections of origin-symmetric convex bodies and their applications

Let V and W be any convex and origin-symmetric bodies in Rn . Assume that for some A ∈LRn→Rn, detA≠0, V is contained in the ellipsoid A−1B2n, where B2n is the unit Euclidean ball. We give a lower bound for the W-radius of sections of A−1V in terms of the spectral radius of A∗A and the expectations of ‖⋅‖V and ‖⋅‖Wo with respect to Haar measure on Sn−1⊂Rn. It is shown that the respective expectations are bounded as n→∞ in many important cases. As an application we offer a new method of evaluation of n-widths of multiplier operators. As an example we establish sharp orders of n-widths of multiplier operators Λ:LpMd→LqMd, 1<q≤2≤p<∞ on compact homogeneous Riemannian manifolds Md. Also, we apply these results to prove the existence of flat polynomials on Md.

Killing vector fields of constant length on compact homogeneous Riemannian manifolds

In this paper we present some structural results on the Lie algebras of transitive isometry groups of a general compact homogenous Riemannian manifold with nontrivial Killing vector fields of constant length.

Compact homogeneous Riemannian manifolds with low coindex of symmetry

We develop a general structure theory for compact homogeneous Riemannian manifolds in relation to the coindex of symmetry.We will then use these results to classify irreducible, simply connected, compact homogeneous Riemannian manifolds whose coindex of symmetry is less than or equal to three. We will also construct many examples which arise from the theory of polars and centrioles in Riemannian symmetric spaces of compact type.

Homogeneous almost normal Riemannian manifolds

In this article, we introduce a newclass of compact homogeneous Riemannian manifolds (M = G/H, µ) almost normal with respect to a transitive Lie group G of isometries for which by definition there exists a G-left-invariant and an H-right-invariant inner product ν such that the canonical projection p: (G, ν) → (G/H, µ) is a Riemannian submersion and the norm | · | of the product ν is at least the bi-invariant Chebyshev normon G defined by the space (M,µ).We prove the following results: Every homogeneous Riemannian manifold is almost normal homogeneous. Every homogeneous almost normal Riemannian manifold is naturally reductive and generalized normal homogeneous. For a homogeneous G-normal Riemannian manifold with simple Lie group G, the unit ball of the norm | · | is a Lowner-John ellipsoid with respect to the unit ball of the Chebyshev norm; an analogous assertion holds for the restrictions of these norms to a Cartan subgroup of the Lie group G. Some unsolved problems are posed.

On δ-homogeneous Riemannian manifolds. II

We continue the study of the δ-homogeneous Riemannian manifolds defined in a more general case by V. N. Berestovskiĭ 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.

On δ-homogeneous Riemannian manifolds

We continue the study of the δ-homogeneous Riemannian manifolds defined in a more general case by V. N. Berestovskiĭ 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.

Inequalities for eigenvalues of Laplacian on domains and compact complex hypersurfaces in complex projective spaces

It is well known that the spectrum of Laplacian on a compact Riemannian manifold M is an important analytic invariant and has important geometric meanings. There are many mathematicians to investigate properties of the spectrum of Laplacian and to estimate the spectrum in term of the other geometric quantities of M . When M is a bounded domain in Euclidean spaces, a compact homogeneous Riemannian manifold, a bounded domain in the standard unit sphere or a compact minimal submanifold in the standard unit sphere, the estimates of the k + 1 -th eigenvalue were given by the first k eigenvalues (see [9], [12], [19], [20], [22], [23], [24] and [25]). In this paper, we shall consider the eigenvalue problem of the Laplacian on compact Riemannian manifolds. First of all, we shall give a general inequality of eigenvalues. As its applications, we study the eigenvalue problem of the Laplacian on a bounded domain in the standard complex projective space C P n ( 4 ) and on a compact complex hypersurface without boundary in C P n ( 4 ) . We shall give an explicit estimate of the k + 1 -th eigenvalue of Laplacian on such objects by its first k eigenvalues.

Estimates on Eigenvalues of Laplacian

In this paper, we study eigenvalues of Laplacian on either a bounded connected domain in an n-dimensional unit sphere Sn(1), or a compact homogeneous Riemannian manifold, or an n-dimensional compact minimal submanifold in an N-dimensional unit sphere SN(1). We estimate the k+1-th eigenvalue by the first k eigenvalues. As a corollary, we obtain an estimate of difference between consecutive eigenvlaues. Our results are sharper than ones of P. C. Yang and Yau [25], Leung [19], Li [20] and Harrel II and Stubbe [12], respectively. From Weyl’s asymptotical formula, we know that our estimates are optimal in the sense of the order of k for eigenvalues of Laplacian on a bounded connected domain in an n-dimensional unit sphere Sn(1).

Compact Homogeneous Hypersurfaces in Hyperbolic Space

In this paper we prove that an n-dimensional compact homogeneous Riemannian manifold isometrically immersed in the hyperbolic space Hn+1 is isometric to a sphere.

The injectivity radius and the fundamental group of compact homogeneous Riemannian manifolds of positive curvature

The injectivity radius and the fundamental group of compact homogeneous Riemannian manifolds of positive curvature

Classification of harmonic mappings of constant energy density into spheres

We classify all eigenmaps and isometric minimal immersions of a flat torus into the unit sphere using the parametrization theorem (cf. [2], [14]) for range-equivalence classes of all eigenmaps of an arbitrary compact homogeneous Riemannian manifold into the unit sphere.

Minimal immersions of compact irreducible homogeneous Riemannian manifolds

Minimal immersions of compact irreducible homogeneous Riemannian manifolds

On the Diameter of Compact Homogeneous Riemannian Manifolds

Let M be a compact Riemannian manifold. The diameter d(M) of M is defined to be the maximum of d(p, q) p, q e M, where d( , ) denotes the distance function on M induced by the Riemannian metric. The main purpose of this paper is to find a positive constant d such that the diameter d(M)^d when the sectional curvature K^l. In this paper we consider the case that the manifold M is homogeneous. In [3] the author proved that d = n/2 if the manifold has a big isotropy subgroup. It has been left to study the case that the isotropy subgroup is finite. Hence we shall mainly study invariant metrics on a Lie group and prove that the number rf>0.23 if the sectional curvature K=£Q (Theorem 5.1).

Compact homogeneous Riemannian manifolds

Compact homogeneous Riemannian manifolds

Compact Homogeneous Riemannian Manifolds with Strictly Positive Curvature

Compact Homogeneous Riemannian Manifolds with Strictly Positive Curvature

The conformal transformation group of a compact homogeneous Riemannian manifold

The conformal transformation group of a compact homogeneous Riemannian manifold

An application of the Morse theory to the topology of Lie-groups

2. The Freudenthal theorem for symmetric spaces Our primary interest is in the classical compact groups; nevertheless, it is essential for our method to consider the larger family of compact symmetric spaces. A compact homogeneous Riemannian manifold M is called symmetric if it admits an 'inverse operation', i.e. if M admits an isometry, keeping a point P e M fixed, and whose differential at P is — 1. These geometric generalizations of the compact groups seem to be the class of spaces to which the Morse theory is most applicable. The reason is that, on such a space, conjugate points have global implications: