Immersions of cohomogeneity one Riemannian manifolds

This work deals with positively curved compact Riemannian manifolds which are acted on by a closed Lie group of isometries whose principal orbits have codimension one and are isotropy irreducible homogeneous spaces. For such manifolds we can show that their universal covering manifold may be isometrically immersed as a hypersurface of revolution in an euclidean space.     

Schouten curvature functions on locally conformally flat Riemannian manifolds

Consider a compact Riemannian manifold (M, g) with metric g and dimension n ≥ 3. The Schouten tensor A _g associated with g is a symmetric (0, 2)-tensor field describing the non-conformally-invariant part of the curvature tensor of g. In this paper, we consider the elementary symmetric functions {σ _k (A _g ), 1 ≤ k ≤ n} of the eigenvalues of A _g with respect to g; we call σ _k (A _g ) the k-th Schouten curvature function. We give an isometric classification for compact locally conformally flat manifolds which satisfy the conditions: A _g is semi-positive definite and σ _k (A _g ) is a nonzero constant for some k ∈ {2, ... , n}. If k = 2, we obtain a classification result under the weaker conditions that σ₂(A _g ) is a non-negative constant and (M ⁿ , g) has nonnegative Ricci curvature. The corresponding result for the case k = 1 is well known. We also give an isometric classification for complete locally conformally flat manifolds with constant scalar curvature and non-negative Ricci curvature. Udo Simon: Partially supported by Chinese-German cooperation projects, DFG PI 158/4-4 and PI 158/4-5, and NSFC.     

Four-dimensional Einstein-like manifolds and curvature homogeneity

The aim of this paper is to classify 4-dimensional Einstein-like manifolds whose Ricci tensor has constant eigenvalues (this being a special kind of curvature homogeneity condition). We give a full classification when the Ricci tensor is of Codazzi type; when the Ricci tensor is cyclic parallel, we classify all such manifolds when not all Ricci curvatures are distinct. In this second case we find a one-parameter family of Riemannian metrics on a Lie groupG as the only possible ones which are irreducible and non-symmetric.     

New examples of Riemannian g.o. manifolds in dimension 7

A Riemannian g.o. manifold is a homogeneous Riemannian manifold (M,g) on which every geodesic is an orbit of a one-parameter group of isometries. It is known that every simply connected Riemannian g.o. manifold of dimension ?5 is naturally reductive. In dimension 6 there are simply connected Riemannian g.o. manifolds which are in no way naturally reductive, and their full classification is known (including compact examples). In dimension 7, just one new example has been known up to now (namely, a Riemannian nilmanifold constructed by C. Gordon). In the present paper we describe compact irreducible 7-dimensional Riemannian g.o. manifolds (together with their “noncompact duals”) which are in no way naturally reductive.     

On the fundamental groups of manifolds with integral curvature bounds

In this paper, we give an upper bound on the growth of π₁(M) for a class of manifolds with integral Ricci curvature bounds. This generalizes the main theorem of [8] to the case where the negative part of Ricci curvature is small in an averaged L¹- sense.Received: 19 July 2004     

Eta-Einstein condition on twistor spaces of odd-dimensional Riemannian manifolds

In this note, we find the conditions on an odd-dimensional Riemannian manifolds under which its twistor space is eta-Einstein. This can be used to yield an Einstein metric on the tangent sphere bundle of any 3-dimensional manifold of positive constant curvature.     

Locally homogeneous Riemannian manifolds and Cartan triples

Ann-dimensional Cartan triple is a triple (g, , ) consisting of a Lie subalgebra g of so(n) (endowed with the Killing form), a linear map : ⁿ g and a bilinear antisymmetric map ²( ⁿ , g), which together satisfy (1.25)–(1.28) of Section 1. LetM _n be the set ofmaximal n-dimensional Cartan triples, and letA _n be thenatural action of the orthogonal group O(n) onM _n (Section 3). One shows that there is a bijective mapping from the set of local isometry classes ofn-dimensional locally homogeneous Riemannian manifolds to the set of orbits ofA _n (Theorem 3.1(a)). Under this bijection, the classes of homogeneous Riemannian manifolds correspond to orbits ofclosed Cartan triples.     

Three-dimensional Lie group actions on compact (4n+3)-dimensional geometric manifolds

The (4n+3)-dimensional sphere S⁴ⁿ⁺³ can be viewed as the boundary of the quaternionic hyperbolic space and the group PSp(n+1,1) of quaternionic hyperbolic isometries extends to a real analytic transitive action on S⁴ⁿ⁺³. We call the pair (PSp(n+1,1),S⁴ⁿ⁺³) a spherical Q C-C geometry. A manifold M locally modelled on this geometry is said to be a spherical Q C-C manifold. We shall classify all pairs (G,M) where G is a three-dimensional connected Lie group which acts smoothly and almost freely on a compact spherical Q C-C manifold M, preserving the geometric structure. As an application, we shall determine all compact 3-pseudo-Sasakian manifolds admitting spherical Q C-C structures.     

Cohomogeneity one Riemannian manifolds of even dimension with strictly positive curvature, I

We classify compact positively curved Riemannian manifolds of even dimension acted on by a semisimple (but not simple) group with a codimension one orbit. Received: 19 September 2000; in final form: 29 October 2001/ Published online: 29 April 2002     

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.     

Lower bounds for the first eigenvalue of the Dirac operator on compact Riemannian manifolds

Some new, improved, curvature depending lower bounds for the first eigenvalue of the Dirac operator on compact Riemannian manifolds are proved. If certain curvature conditions are satisfied, then these lower bounds are also useful in cases where the scalar curvature has zeros or attains negative values. This implies stronger vanishing theorems for harmonic spinors.     

Classification of generalized normal homogeneous Riemannian manifolds of positive Euler characteristic

The authors give a short survey of previous results on generalized normal homogeneous (δ-homogeneous, in other terms) Riemannian manifolds, forming a new proper subclass of geodesic orbit spaces with nonnegative sectional curvature, which properly includes the class of all normal homogeneous Riemannian manifolds. As a continuation and an application of these results, they prove that the family of all compact simply connected indecomposable generalized normal homogeneous Riemannian manifolds with positive Euler characteristic, which are not normal homogeneous, consists exactly of all generalized flag manifolds Sp(l)/U(1)⋅Sp(l−1)=CP2l−1, l?2, supplied with invariant Riemannian metrics of positive sectional curvature with the pinching constants (the ratio of the minimal sectional curvature to the maximal one) in the open interval (1/16,1/4). This implies very unusual geometric properties of the adjoint representation of Sp(l), l?2. Some unsolved questions are suggested.     

The Gagliardo-Nirenberg inequalities and manifolds of non-negative Ricci curvature

This article is devoted to show that complete non-compact Riemannian manifolds with non-negative Ricci curvature of dimension greater than or equal to two in which some Gagliardo-Nirenberg type inequality holds are not very far from the Euclidean space.     

Symmetric hypersurfaces in Riemannian manifolds contracting to Lie-groups by their mean curvature

This paper concerns the deformation by mean curvature of hypersurfaces M in Riemannian spaces Ñ that are invariant under a subgroup of the isometry-group on Ñ. We show that the hypersurfaces contract to this subgroup, if the cross-section satisfies a strong convexity assumption.This forms part of the authors doctoral thesis and was carried out while the author was supported by a scholarship of the Graduiertenkolleg für Geometrie und Mathematische Physik.     

Harmonic maps from closed Riemannian manifolds with positive scalar curvature

It is well known there is no non-constant harmonic map from a closed Riemannian manifold of positive Ricci curvature to a complete Riemannian manifold with non-positive sectional curvature. By reducing the assumption on the Ricci curvature to one on the scalar curvature, such vanishing theorem cannot hold in general. This raises the question: “What information can we obtain from the existence of non-constant harmonic map?” This paper gives answer to this problem; the results obtained are optimal.     

Classification of Riemannian manifolds in nonlinear potential theory

The classification theory of Riemann surfaces is generalized to Riemanniann-manifolds in the conformally invariant case. This leads to the study of the existence ofA-harmonic functions of typen with various properties and to an extension of the definition of the classical notions with inclusionsO _G O _HP O _HB O _HD . In the classical case the properness of the inclusions were proved rather late, in the 50's by Ahlfors and Tôki. Our main objective is to show that such inclusions are proper also in the generalized case.This research was supported in part by grants from the Academy of Finland and the U.S. National Science Foundation (NSF DMS 9003438).