On the volume of the intersection of two geodesic balls

The first author and D. Kunszenti-Kovács (2010) [1] proved that if the volume of the intersection of three geodesic balls of a complete connected Riemannian manifold depends only on the center-center distances and the radii of the balls, then the manifold is one of the simply connected spaces of constant curvature. In this paper, we study the geometrical consequences of the analogous condition for pairs of geodesic balls. We show that in a complete, connected and simply connected Riemannian manifold, the volume of the intersection of two small geodesic balls depends only on the distance between the centers and the radii if and only if the space is harmonic. It is also shown that if in a Riemannian manifold the volume of the intersection of two small geodesic balls of equal radii depends only on the distance between the centers and the common value of the radii, then the space is Einstein, and if we assume in addition that the space is symmetric, then it must be Osserman and hence two-point homogeneous.     

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.     

Homogeneous geodesics in solvable Lie groups

After the second author and J. Szenthe [10] proved that every homogeneous Riemannian manifold admits a homogeneous geodesic, several authors studied the set of all homogeneous geodesics in various homogeneous spaces. In this paper, we consider special examples of homogeneous spaces of solvable type of arbitrary odd dimension given in [1] and [7] and we show that their sets of homogeneous geodesics have an interesting structure, closely connected to the notion of Hadamard matrices.     

On the Existence of Homogeneous Geodesics in Homogeneous Riemannian Manifolds

It is proved that every homogeneous Riemannian manifold admits a geodesic which is an orbit of a one-parameter group of isometries.     

Singular Riemannian foliations on simply connected spaces

A singular foliation on a complete Riemannian manifold is said to be Riemannian if each geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets. The singular foliation is said to admit sections if each regular point is contained in a totally geodesic complete immersed submanifold that meets every leaf orthogonally and whose dimension is the codimension of the regular leaves. A typical example of such a singular foliation is the partition by orbits of a polar action, e.g. the orbits of the adjoint action of a compact Lie group on itself.We prove that a singular Riemannian foliation with compact leaves that admits sections on a simply connected space has no exceptional leaves, i.e., each regular leaf has trivial normal holonomy. We also prove that there exists a convex fundamental domain in each section of the foliation and in particular that the space of leaves is a convex Coxeter orbifold.     

On causality and closed geodesics of compact Lorentzian manifolds and static spacetimes

Some results related to the causality of compact Lorentzian manifolds are proven: (1) any compact Lorentzian manifold which admits a timelike conformal vector field is totally vicious, and (2) a compact Lorentzian manifold covered regularly by a globally hyperbolic spacetime admits a timelike closed geodesic, if some natural topological assumptions (fulfilled, for example, if one of the conjugacy classes of deck transformations containing a closed timelike curve is finite) hold. As a consequence, any compact Lorentzian manifold conformal to a static spacetime is geodesically connected by causal geodesics, and admits a timelike closed geodesic.     

Natural almost contact structures and their 3D homogeneous models

We consider a natural condition determining a large class of almost contact metric structures. We study their geometry, emphasizing that this class shares several properties with contact metric manifolds. We then give a complete classification of left‐invariant examples on three‐dimensional Lie groups, and show that any simply connected homogeneous Riemannian three‐manifold admits a natural almost contact structure having g as a compatible metric. Moreover, we investigate left‐invariant CR structures corresponding to natural almost contact metric structures.     

The Ricci flow in a class of solvmanifolds

In this paper, we study the Ricci flow of solvmanifolds whose Lie algebra has an abelian ideal of codimension one, by using the bracket flow. We prove that solutions to the Ricci flow are immortal, the ω-limit of bracket flow solutions is a single point, and that for any sequence of times there exists a subsequence in which the Ricci flow converges, in the pointed topology, to a manifold which is locally isometric to a flat manifold. We give a functional which is non-increasing along a normalized bracket flow that will allow us to prove that given a sequence of times, one can extract a subsequence converging to an algebraic soliton, and to determine which of these limits are flat. Finally, we use these results to prove that if a Lie group in this class admits a Riemannian metric of negative sectional curvature, then the curvature of any Ricci flow solution will become negative in finite time.     

Submersions and curves of constant geodesic curvature

Considering Riemannian submersions, we find necessary and sufficient conditions for when sub‐Riemannian normal geodesics project to curves of constant first geodesic curvature or constant first and vanishing second geodesic curvature. We describe a canonical extension of the sub‐Riemannian metric and study geometric properties of the obtained Riemannian manifold. This work contains several examples illustrating the results.     

On fat sub-Riemannian symmetric spaces in codimension three

A sub-Riemannian manifold is a differentiable manifold together with a smooth distribution which is equipped with a Riemannian metric. In this paper we attempt to study sub-Riemannian symmetric spaces (i.e., homogeneous sub-Riemannian manifolds admitting an involutive sub-Riemannian isometry at all points which is a central symmetry when restricted to the distribution) where the associated distribution is a codimension three fat distribution. We obtain a restricted classification theorem in dimension seven and we also construct a class of examples of quaternionic type in varying dimension.     

Homogeneous two-manifolds with an invariant two-form

We study a 2-dimensional manifold that admits a homogeneous action of a 3-dimensional Lie group G, and has a 2-form invariant under G. We show that such a manifold can be realized as a surface in the affine 3-space, and list such realizations.      

Some Finsler spaces with homogeneous geodesics

A geodesic in a homogeneous Finsler space is called a homogeneous geodesic if it is an orbit of a one‐parameter subgroup of G . A homogeneous Finsler space is called Finsler g.o. space if its all geodesics are homogeneous. Recently, the author studied Finsler g.o. spaces and generalized some geometric results on Riemannian g.o. spaces to the Finslerian setting. In the present paper, we investigate homogeneous geodesics in homogeneous spaces, and obtain the sufficient and necessary condition for an space to be a g.o. space. As an application, we get a series of new examples of Finsler g.o. spaces.     

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.     

Obstructions to toric integrable geodesic flows in dimension 3

The geodesic flow of a Riemannian metric on a compact manifold Q is said to be toric integrable if it is completely integrable and the first integrals of motion generate a homogeneous torus action on the punctured cotangent bundle T ^* Q\Q. If the geodesic flow is toric integrable, the cosphere bundle admits the structure of a contact toric manifold. By comparing the Betti numbers of contact toric manifolds and cosphere bundles, we are able to provide necessary conditions for the geodesic flow on a compact, connected 3-dimensional Riemannian manifold to be toric integrable.Mathematics Subject Classifications (2000): primary 53D25; secondary 53D10     

On geodesic mappings of equidistant generalized Riemannian spaces

In this paper geodesic mappings of equidistant generalized Riemannian spaces are discussed. It is proved that each equidistant generalized Riemannian space of basic type admits non-trivial geodesic mapping with preserved equidistant congruence. Especially, there exists non-trivial geodesic mapping of equidistant generalized Riemannian space onto equidistant Riemannian space. An example of geodesic mapping of an equidistant generalized Riemannian spaces is presented.     

Ricci soliton homogeneous nilmanifolds

We study a notion weakening the Einstein condition on a left invariant Riemannian metric g on a nilpotent Lie groupN. We consider those metrics satisfying Ric for some and some derivationD of the Lie algebra ofN, where Ric denotes the Ricci operator of . This condition is equivalent to the metric g to be a Ricci soliton. We prove that a Ricci soliton left invariant metric on N is unique up to isometry and scaling. The following characterization is also given: (N,g) is a Ricci soliton if and only if (N,g) admits a metric standard solvable extension whose corresponding standard solvmanifold is Einstein. This gives several families of new examples of Ricci solitons. By a variational approach, we furthermore show that the Ricci soliton homogeneous nilmanifolds (N,g) are precisely the critical points of a natural functional defined on a vector space which contains all the homogeneous nilmanifolds of a given dimension as a real algebraic set. Received August 24, 1999 / Revised October 2, 2000 / Published online February 5, 2001     

Volume comparison and the σk-Yamabe problem

In this paper we study the problem of finding a conformal metric with the property that the kth elementary symmetric polynomial of the eigenvalues of its Weyl-Schouten tensor is constant. A new conformal invariant involving maximal volumes is defined, and this invariant is then used in several cases to prove existence of a solution, and compactness of the space of solutions (provided the conformal class admits an admissible metric). In particular, the problem is completely solved in dimension four, and in dimension three if the manifold is not simply connected.     

Convexity and Some Geometric Properties

The main goal of this paper is to present results of existence and nonexistence of convex functions on Riemannian manifolds, and in the case of the existence, we associate such functions to the geometry of the manifold. Precisely, we prove that the conservativity of the geodesic flow on a Riemannian manifold with infinite volume is an obstruction to the existence of convex functions. Next, we present a geometric condition that ensures the existence of (strictly) convex functions on a particular class of complete manifolds, and we use this fact to construct a manifold whose sectional curvature assumes any real value greater than a negative constant and admits a strictly convex function. In the last result, we relate the geometry of a Riemannian manifold of positive sectional curvature with the set of minimum points of a convex function defined on the manifold.     

Polar actions with a fixed point

We prove a criterion for an isometric action of a Lie group on a Riemannian manifold to be polar. From this criterion, it follows that an action with a fixed point is polar if and only if the slice representation at the fixed point is polar and the section is the tangent space of an embedded totally geodesic submanifold. We apply this to obtain a classification of polar actions with a fixed point on symmetric spaces.     

On a class of twistorial maps

We show that a natural class of twistorial maps gives a pattern for apparently different geometric maps, such as, (1,1)-geodesic immersions from (1,2)-symplectic almost Hermitian manifolds and pseudo horizontally conformal submersions with totally geodesic fibres for which the associated almost CR-structure is integrable. Along the way, we construct for each constant curvature Riemannian manifold (M,g), of dimension m, a family of twistor spaces such that Zr(M) parametrizes naturally the set of pairs (P,J), where P is a totally geodesic submanifold of (M,g), of codimension 2r, and J is an orthogonal complex structure on the normal bundle of P which is parallel with respect to the normal connection.