We exhibit a family of generalized plane wave manifolds of signature (2,2). The geodesics in these manifolds extend for infinite time (i.e. they are complete), they are spacelike and timelike Jordan Osserman, and they are spacelike and timelike Jordan Ivanov-Petrova. Some are irreducible symmetric spaces. Some are homogeneous spaces but not symmetric. Some are 1-curvature homogeneous but not homogeneous. All are 0-modeled on the same irreducible symmetric space. We determine the Killing vector fields for these manifolds.
We study a family of 3-dimensional Lorentz manifolds. Some members of the family are 0-curvature homogeneous, 1-affine curvature homogeneous, but not 1-curvature homogeneous. Some are 1-curvature homogeneous but not 2-curvature homogeneous. All are 0-modeled on indecomposible local symmetric spaces. Some of the members of the family are geodesically complete, others are not. All have vanishing scalar invariants.
We construct a family of balanced signature pseudo-Riemannian manifolds, which arise as hypersurfaces in flat space, that are curvature homogeneous, that are modeled on a symmetric space, and that are not locally homogeneous.
We exhibit 3 families of complete curvature homogeneous pseudo-Riemannian manifolds which are modeled on irreducible symmetric spaces and which are not locally homogeneous. All of the manifolds have nilpotent Jacobi operators; some of the manifolds are, in addition, Jordan Osserman and Jordan Ivanov-Petrova.
Let s be at least 2. We construct Ricci flat pseudo-Riemannian manifolds of signature (2s,s) which are not locally homogeneous but whose curvature tensors never the less exhibit a number of important symmetry properties. They are curvature homogeneous; their curvature tensor is modeled on that of a local symmetric space. They are spacelike Jordan Osserman with a Jacobi operator which is nilpotent of order 3; they are not timelike Jordan Osserman. They are k-spacelike higher order Jordan Osserman for $2\le k\le s$; they are k-timelike higher order Jordan Osserman for $s+2\le k\le 2s$, and they …
I apply the algebraic classification of self-adjoint endomorphisms of ${\bf R}^{2,2}$ provided by their Jordan canonical form to the Ricci curvature tensor of four-dimensional neutral manifolds and relate this classification to an algebraic classification of the Ricci curvature spinor. These results parallel similar results well known in four-dimensional Lorentzian geometry. The classification is summarized in Table 2 at the end of the paper.
The main aim of this article is to calculate explicit formula for S-curvature in homogeneous generalized m-Kropina metric. Further, we also deduce mean Berwald curvature for homogeneous generalized m-Kropina metric from S-curvature.
For k at least 2, we exhibit complete k-curvature homogeneous neutral signature pseudo-Riemannian manifolds which are not locally affine homogeneous (and hence not locally homogeneous). The curvature tensor of these manifolds is modeled on that of an indecomposible symmetric space. All the local scalar Weyl curvature invariants of these manifolds vanish.
For any k which is at least 2, we exhibit complete k-curvature homogeneous neutral signature pseudo-Riemannian manifolds which are not k+1-affine curvature homogeneous, and hence not locally homogeneous. All the local scalar Weyl invariants of these manifolds vanish. These manifolds are Ricci flat, Osserman, and Ivanov-Petrova.
We show that generalized plane wave manifolds are complete, strongly geodesically convex, Osserman, Szabo, and Ivanov-Petrova. We show their holonomy groups are nilpotent and that all the local Weyl scalar invariants of these manifolds vanish. We construct isometry invariants on certain families of these manifolds which are not of Weyl type. Given k, we exhibit manifolds of this type which are k-curvature homogeneous but not locally homogeneous. We also construct a manifold which is weakly 1-curvature homogeneous but not 1-curvature homogeneous.
We classify homogeneous reversible Finsler metrics with positive Flag curvature. We show that if G/H admits a G invariant reversible Finsler metric with positive Flag curvature, then up to a few low dimensional spaces, it also admits a G invariant Riemannian metric with positive sectional curvature. For the exceptions, we do not know if they admit homogeneous Finsler metrics with positive Flag curvature.
We study the isometry groups and Killing vector fields of a family of pseudo-Riemannian metrics on Euclidean space which have neutral signature (3+2p,3+2p). All are p+2 curvature homogeneous, all have vanishing Weyl scalar invariants, all are geodesically complete, and all are 0-curvature modeled on an indecomposible symmetric space. Some of these manifolds are not p+3 curvature homogeneous. Some are homogeneous but not symmetric.
For Fano homogeneous toric bundles, we obtain a formula of the greatest lower bound on Ricci curvature. We also give a criteria for the ampleness of a kind of line bundles over general homogeneous toric bundles.
The study of curvature properties of homogeneous Finsler spaces with $(α, β)$-metrics is one of the central problems in Riemann-Finsler geometry. In this paper, we consider homogeneous Finsler spaces with square metric and Randers change of square metric. First, we derive the explicit formulae for Ricci curvature of homogeneous Finsler spaces with these metrics. Next, we find a necessary and sufficient condition under which a homogeneous Finsler space with either of these metrics is of vanishing $S$-curvature. The formulae for Ricci curvature of homogeneous Finsler spaces with square metric an…
In this paper, we study the positive cross curvature flow on locally homogeneous 3-manifolds. We describe the long time behavior of these flows. We combine this with earlier results concerning the asymptotic behavior of the negative cross curvature flow to describe the two sided behavior of maximal solutions of the cross curvature flow on locally homogeneous 3-manifolds. We show that, typically, the positive cross curvature flow on locally homogeneous 3-manifold produce an Heisenberg type sub-Riemannian geometry.
In this paper we classify complete surfaces of constant mean curvature whose Gaussian curvature does not change sign in a simply connected homogeneous manifold with a 4-dimensional isometry group.
In this paper, we first deduce a formula of S-curvature of homogeneous Finsler spaces in terms of Killing vector fields. Then we prove that a homogeneous Finsler space has isotropic S-curvature if and only if it has vanishing S-curvature. In the special case that the homogeneous Finsler space is a Randers space, we give an explicit formula which coincides with the previous formula obtained by the second author using other methods.
We prove local existence of complex-valued harmonic morphisms from any Riemannian homogeneous spaces of positive curvature, except the Berger space Sp(2)/SU(2).
Compact flat surfaces of homogeneous Riemannian 3-manifolds with isometry group of dimension 4 are classified. Non-existence results for compact constant Gauss curvature surfaces in these 3-manifolds are established.
We prove that a four-dimensional Lorentzian manifold that is curvature homogeneous of order 3, or $\CH_3$ for short, is necessarily locally homogeneous. We also exhibit and classify four-dimensional Lorentzian, $\CH_2$ manifolds that are not homogeneous.
In this paper, first, we give an explicit formula for the flag curvature of a homogeneous Finsler space with generalized $m$-Kropina metric. Then, we show that, under a mild condition, the two definitions of naturally reductive homogeneous Finsler space are equivalent for afore said metric. Finally, we study the flag curvature of naturally reductive homogeneous Finsler spaces with generalized $m$-Kropina metric.
The main purpose of this paper is to determine the admissible forms of the sectional curvature operator on a three-dimensional locally homogeneous Lorentzian manifolds.
We characterize cohomogeneity one manifolds and homogeneous spaces with a compact Lie group action admitting an invariant metric with positive scalar curvature.
We extend the definition of curvature homogeneity of type (1,3) to include the possibility that there is a homothety between any two points of a manifold preserving the first r covariant derivatives of the curvature operator simultaneously; we call this strong curvature homogeneity of type (1,3) up to order r. We characterize these properties in terms of model spaces. In addition, we also present two families of three-dimensional Lorentzian metrics on Euclidean space to exhibit the behavior of this property. The first example is curvature homogeneous of type (1,3) of all orders, but is not loc…
The study of curvature properties of homogeneous Finsler spaces with $(α, β)$-metrics is one of the central problems in Riemann-Finsler geometry. In the present paper, the existence of invariant vector fields on homogeneous Finsler spaces with square $(α, β)$-metric and Randers changed square $(α, β)$-metric is proved. Further, an explicit formula for $S$-curvature of these $(α, β)$-metrics is established. Finally, using the formula of $S$-curvature, the mean Berwald curvature of afore said $(α, β)$-metrics is calculated.
We prove that a 2n-dimensional compact homogeneous nearly Kahler manifold with strictly positive sectional curvature is isometric to CP^{n}, equipped with the symmetric Fubini-Study metric or with the standard Sp(m)-homogeneous metric, n =2m-1, or to S^{6} as Riemannian manifold with constant sectional curvature. This is a positive answer for a revised version of a conjecture given by Gray.
In this paper, first we derive an explicit formula for the flag curvature of a homogeneous Finsler space with infinite series $(α, β)$-metric and exponential metric. Next, we deduce it for naturally reductive homogeneous Finsler space with the above mentioned metrics.
This is a survey paper on our recent works concerning the classification of positively curved homogeneous Finsler spaces, and some related topics. At the final part, we present some open problems in this field.
Consider a compact Lie group $G$ and a closed Lie subgroup $H<G$. Let $\mathcal M$ be the set of $G$-invariant Riemannian metrics on the homogeneous space $M=G/H$. By studying variational properties of the scalar curvature functional on $\mathcal M$, we obtain an existence theorem for solutions to the prescribed Ricci curvature problem on $M$. To illustrate the applicability of this result, we explore cases where $M$ is a generalised Wallach space and a generalised flag manifold.
Invariant complex structures on the homogeneous manifold $U(n+1)/U(n)\times U(p+1)/U(p)$ are reseached. Extreme values of sectional curvature of Hermitian metrics on this manifold are found.
J. Streets and G. Tian recently introduced symplectic curvature flow, a geometric flow on almost Kähler manifolds generalising Kähler-Ricci flow. The present article gives examples of explicit solutions to this flow of non-Kähler structures on several nilmanifolds and on twistor fibrations over hyperbolic space studied by J. Fine and D. Panov. The latter lead to examples of non-Kähler static solutions of symplectic curvature flow which can be seen as analogues of Kähler-Einstein manifolds in Kähler-Ricci flow.
We provide new examples of manifolds which admit a Riemannian metric with sectional curvature nonnegative, and strictly positive at one point. Our examples include the unit tangent bundles of $CP^n$, $HP^n$ and $CaP^2$, and a family of lens space bundles over $CP^n$. All new examples are consequences of a general sufficient condition for a homogeneous fiber bundle over a homogeneous space to admit such a metric.
In this paper, we use the flag curvature formula for homogeneous Finsler spaces in our previous work to classify odd dimensional smooth coset spaces admitting positively curved reversible homogeneous Finsler metrics. We will show that the most features of L. Bérard-Bergery's classification results for odd dimensional positively curved Riemannian homogeneous spaces can be generalized to reversible Finsler spaces.
We study a version of the Hermitian curvature flow on compact homogeneous complex manifolds. We prove that the solution has a finite exstinction time $T>0$ and we analyze its behaviour when $t\to T$. We also determine the invariant static metrics and we study the convergence of the normalized flow to one of them.
In this article we review the recent results about the flag curvature of invariant Randers metrics on homogeneous manifolds and by using a counter example we show that the formula which obtained for the flag curvature of these metrics is incorrect. Then we give an explicit formula for the flag curvature of invariant Randers metrics on the naturally reductive homogeneous manifolds $(G/H,g)$, where the Randers metric induced by the invariant Riemannian metric $g$ and an invariant vector field $\tilde{X}$ which is parallel with $g$.
We prove that two spheres of the same constant mean curvature in an arbitrary homogeneous three-manifold only differ by an ambient isometry, and we determine the values of the mean curvature for which such spheres exist. This gives a complete classification of immersed constant mean curvature spheres in three-dimensional homogeneous manifolds.
In this paper, we present the classification of all possible signatures of the Ricci curvature of left-invariant Riemannian metrics on 4-dimensional Lie groups and discuss some related questions.
Let $R$ be an algebraic curvature tensor on a vector space of signature $(p,q)$ defining a spacelike Jordan Osserman Jacobi operator $\JJ_R$. We show that the eigenvalues of $\JJ_R$ are real and that $\JJ_R$ is diagonalizable if $p<q$.
In this paper we construct a family of examples of self-dual Einstain metrics of neutral signature, which are not Ricci flat, nor locally homogenous. Curvature of these manifolds is studied in details. These are obtained by the para-quaternionic reduction. We compare our examples with the orbifolds $\oo$ given by Galicki and Lawson, for which some new properties are also established. Particularly, the sign and the pinching of their sectional curvatures are studied.
Duncan and Ihrig (1993) gave a classification of the flat homogeneous spaces of metric signature (m,2), provided that a certain condition on the development image of these spaces holds. In this note we show that this condition can be dropped, so that Duncan and Ihrig's classification is in fact the full classification for signature (m,2).
A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to be topologically formal. Nonetheless, we show that among the homogeneous Riemannian metrics of positive sectional curvature a geometrically formal metric is either symmetric, or a metric on a rational homology sphere.
We present the classical Wagner construction from 1935 of the curvature tensor for completely nonholonomic manifolds in both invariant and coordinate way. The starting point is the Shouten curvature tensor for nonholonomic connection introduced by Vranceanu and Shouten. We illustrate the construction on two mechanical examples: the case of a homogeneous disc rolling without sliding on a horizontal plane and the case of a homogeneous ball rolling without sliding on a fixed sphere. In the second case we study the conditions on the ratio of diameters of the ball and the sphere to obtain a flat sp…
In this paper, we study the generalized Abreu equation on a Delzant ploytope $Δ\subset \mathbb{R}^2$ and prove the existence of the constant scalar metrics of homogeneous toric bundles under the assumption of an appropriate stability.
In this paper, we prove two rigidity results for non-positively curved homogeneous Finsler metrics. Our first main result yields an extension of Hu-Deng's well-known result proven for the Randers metrics. Indeed, we prove that every connected homogeneous Finsler space with non-positive flag curvature and isotropic S-curvature is Riemannian or locally Minkowskian. We extend the Szabó's rigidity theorem for Berwald surfaces and show that homogeneous isotropic Berwald metrics with non-positive flag curvature are Riemannian or locally Minkowskian. We prove that a homogeneous $(α, β)$-metrics has i…
A Riemannian manifold is called harmonic if its volume density function expressed in polar coordinates centered at any point is radial. Flat and rank-one symmetric spaces are harmonic. The converse (the Lichnerowicz Conjecture) is true for manifolds of nonnegative scalar curvature and for some other classes of manifolds, but is not true in general: there exists a family of homogeneous harmonic spaces, the Damek-Ricci spaces, containing noncompact rank-one symmetric spaces, as well as infinitely many nonsymmetric examples. We prove that a harmonic homogeneous manifold of nonpositive curvature i…
We give a complete classification of the immersed constant mean curvature spheres in a three-sphere with an arbitrary homogenous metric, by proving that for each $H\in\mathbb{R}$, there exists a constant mean curvature $H$-sphere in the space that is unique up to an ambient isometry.
We find out upper bounds for the first eigenvalue of the stability operator for compact constant mean curvature surfaces immersed into certain 3-dimensional Riemannian spaces, in particular into homogeneous 3-manifolds. As an application we derive some consequences for strongly stable surfaces in such ambient spaces. Moreover, we also get a characterization of Hopf tori in certain Berger spheres.
We completely describe the signatures of the Ricci curvature of left-invariant Riemannian metrics on arbitrary real nilpotent Lie groups. The main idea in the proof is to exploit a link between the kernel of the Ricci endomorphism and closed orbits in a certain representation of the general linear group, which we prove using the `real GIT' framework for the Ricci curvature of nilmanifolds.
We show that every n-dimensional locally homogeneous pp-wave is a plane wave, provided it is indecomposable and its curvature operator, when acting on $2$-forms, has rank greater than one. As a consequence we obtain that indecomposable, Ricci-flat locally homogeneous pp-waves are plane waves. This generalises a classical result by Jordan, Ehlers and Kundt in dimension 4. Several examples show that our assumptions on indecomposability and the rank of the curvature are essential.
We exhibit Walker manifolds of signature (2,2) with various commutativity properties for the Ricci operator, the skew-symmetric curvature operator, and the Jacobi operator. If the Walker metric is a Riemannian extension of an underlying affine structure A, these properties are related to the Ricci tensor of A.
In this paper we investigate the behavior of three-dimensional homogeneous solutions of the cross curvature flow using Riemannian groupoids. The Riemannian groupoid technique, introduced by John Lott, allows us to investigate the long term behavior of collapsing solutions of the flow, producing soliton solutions in the limit. We also review Lott's results on the long term behavior of three-dimensional homogeneous solutions of Ricci flow, demonstrating the coordinates we choose and reviewing the groupoid technique. We find cross curvature soliton metrics on Sol and Nil, and show that the cross …
Given compact Lie groups H\subset G, we study the space of G-invariant metrics on G/H with nonnegative sectional curvature. For an intermediate subgroup K between H and G, we derive conditions under which enlarging the Lie algebra of K maintains nonnegative curvature on G/H. Such an enlarging is possible if (K,H) is a symmetric pair, which yields many new examples of nonnegatively curved homogeneous metrics. We provide other examples of spaces G/H with unexpectedly large families of nonnegatively curved homogeneous metrics.
In this paper, we use the technique of Finslerian submersion to deduce a flag curvature formula for homogeneous Finsler spaces. Based on this formula, we give a complete classification of even-dimensional smooth coset spaces $G/H$ admitting $G$-invariant Finsler metrics with positive flag curvature. It turns out that the classification list coincides with that of the even dimensional homogeneous Riemannian manifolds with positive sectional curvature obtained by N.R. Wallach. We also find out all the coset spaces admitting invariant non-Riemannian Finsler metrics with positive flag curvature.
We prove that a homogeneous Finsler sphere with constant flag curvature $K\equiv1$ and a prime closed geodesic of length $2π$ must be Riemannian. This observation provides the evidence for the non-existence of homogeneous Bryant spheres. It also helps us propose an alternative approach proving that a geodesic orbit Finsler sphere with $K\equiv1$ must be Randers. Then we discuss the behavior of geodesics on a homogeneous Finsler sphere with $K\equiv1$. We prove that many geodesic properties for homogeneous Randers spheres with $K\equiv1$ can be generalized to the non-Randers case.
We study the spectral geometry of the Riemann curvature tensor for Pseudo-Riemannian manifolds and provide some examples illustrating the phenomena which can arise in the higher signature setting. Dedication: This paper is dedicated to the memory of our colleague Prof.G. Tsagas who studied the spectral geometry of Laplacian.
