The Hijazi inequality on conformally parabolic manifolds

We prove the Hijazi inequality, an estimate for Dirac eigenvalues, for complete manifolds of finite volume. Under some additional assumptions on the dimension and the scalar curvature, this inequality is also valid for elements of the essential spectrum. This allows to prove the conformal version of the Hijazi inequality on conformally parabolic manifolds if the spin analog to the Yamabe invariant is positive.     

Almost Hermitian structures and quaternionic geometries

Gray and Hervella gave a classification of almost Hermitian structures (g,I) into 16 classes. We systematically study the interaction between these classes when one has an almost hyper-Hermitian structure (g,I,J,K). In general dimension we find at most 167 different almost hyper-Hermitian structures. In particular, we obtain a number of relations that give hyper-Kähler or locally conformal hyper-Kähler structures, thus generalising a result of Hitchin. We also study the types of almost quaternion-Hermitian geometries that arise and tabulate the results.     

The Weyl functional near the Yamabe invariant

For a compact manifold M ofdim M=n≥4, we study two conformal invariants of a conformal class C on M. These are the Yamabe constant Y_C(M) and the L^n/2-norm W_C(M) of the Weyl curvature. We prove that for any manifold M there exists a conformal class C such that the Yamabe constant Y_C(M) is arbitrarily close to the Yamabe invariant Y(M), and, at the same time, the constant W_C(M) is arbitrarily large. We study the image of the mapYW:C→(Y_C(M), W_C(M))∈R ² near the line {(Y(M), w)|w∈R}. We also apply our results to certain classes of 4-manifolds, in particular, minimal compact Kähler surfaces of Kodaira dimension 0, 1 or 2.     

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.     

Surgery and equivariant Yamabe invariant

We consider the equivariant Yamabe problem, i.e., the Yamabe problem on the space of G-invariant metrics for a compact Lie group G. The G-Yamabe invariant is analogously defined as the supremum of the constant scalar curvatures of unit volume G-invariant metrics minimizing the total scalar curvature functional in their G-invariant conformal subclasses. We prove a formula about how the G-Yamabe invariant changes under the surgery of codimension 3 or more, and compute some G-Yamabe invariants.     

Manifolds of positive scalar curvature and conformal cobordism theory

For a supergoup , we study closed -manifolds with positive conformal classes. We use the relative Yamabe invariant from [2] to define the conformal cobordism relation on the category of such manifolds. We prove that the corresponding conformal cobordism groups are isomorphic to the cobordism groups defined by Stolz in [19]. As a corollary, we show that the conformal concordance relation on positive conformal classes coincides with the standard concordance relation on positive scalar curvature metrics. Our main technical tools come from analysis and conformal geometry. Received: 22 August 2000 / Published online: 5 September 2002     

Almost Kähler 4-dimensional Lie groups with J-invariant Ricci tensor

The J-invariance of the Ricci tensor is a natural weakening of the Einstein condition in almost Hermitian geometry. The aim of this paper is to determine left-invariant strictly almost Kähler structures (g,J,Ω) on real 4-dimensional Lie groups such that the Ricci tensor is J-invariant. We prove that all these Lie groups are isometric (up to homothety) to the (unique) 4-dimensional proper 3-symmetric space.     

Some Aspects on the Geometry of the Tangent Bundles and Tangent Sphere Bundles of a Riemannian Manifold

In this paper we study a Riemannian metric on the tangent bundle T(M) of a Riemannian manifold M which generalizes Sasaki metric and Cheeger–Gromoll metric and a compatible almost complex structure which confers a structure of locally conformal almost K?hlerian manifold to T(M) together with the metric. This is the natural generalization of the well known almost K?hlerian structure on T(M). We found conditions under which T(M) is almost K?hlerian, locally conformal K?hlerian or K?hlerian or when T(M) has constant sectional curvature or constant scalar curvature. Then we will restrict to the unit tangent bundle and we find an isometry with the tangent sphere bundle (not necessary unitary) endowed with the restriction of the Sasaki metric from T(M). Moreover, we found that this map preserves also the natural contact structures obtained from the almost Hermitian ambient structures on the unit tangent bundle and the tangent sphere bundle, respectively. This work was also partially supported by Grant CEEX 5883/2006–2008, ANCS, Romania.     

On Einstein Hermitian manifolds

We show that every compact Einstein Hermitian surface with constant *–scalar curvature is a K?hler surface. In contrast to the 4-dimensional case, it is shown that there exists a compact Einstein Hermitian (4n + 2)-dimensional manifold with constant *–scalar curvature which is not K?hler. This study is supported by Kangwon National University.     

Almost Hermitian 4-manifolds of pointwise constant anti-holomorphic sectional curvature

We show that a 4-dimensional almost Hermitian manifold (M, J, g) is of pointwise constant anti-holomorphic sectional curvature if and only if (M, J, g) is self-dual with J-invariant Ricci tensor and K₁₂₁₂ = 0, where K is the complexification of the Riemannian curvature tensor.     

On the chromatic polynomial of a graph

Let P(G,λ) be the chromatic polynomial of a graph G with n vertices, independence number α and clique number ω. We show that for every λ≥n, ()^α≤≤ () ^{n −ω}. We characterize the graphs that yield the lower bound or the upper bound.?These results give new bounds on the mean colour number μ(G) of G: n− (n−ω)() ^{n −ω}≤μ(G)≤n−α() ^α. Received: December 12, 2000 / Accepted: October 18, 2001?Published online February 14, 2002     

Some remarks on Legendrian rectifiable currents

A rectifiable current of dimension n−1 in the sphere bundle Sℝⁿ≃ℝⁿ×S ^{n −1} for euclidean space is Legendrian if it annihilates the contact 1-form α (i.e. T(α∧φ)=0 for all forms φ of degree n−2). Such a current may be naturally associated to any convex set or to any singular real analytic variety, and induces the curvature measures of such a set. We prove that the projection to ℝⁿ of a carrier of a general such T is C ²-rectifiable in the sense of Anzellotti–Serapioni. We deduce that the boundary of a set with positive reach, as well as its singular skeleta, are C ²-rectifiable. In case ∂T= 0 we prove also that the curvature measures associated to T satisfy the analogues of the classical variational formulas for curvature integrals. It follows that such formulas are valid for the curvature measures of subsets of space forms. Received: 3 December 1997/ Revised version: 25 May 1998     

Studies of Some Curvature Operators in a Neighborhood of an Asymptotically Hyperbolic Einstein Manifold

On an asymptotically hyperbolic Einstein manifold (M,g₀) for which the Yamabe invariant of the conformal structure on the boundary at infinity is nonnegative, we show that the operators of Ricci curvature, and of Einstein curvature, are locally invertible in a neighborhood of the metric g₀. We deduce in the C^∞ case that the image of the Riemann-Christoffel curvature operator is a submanifold in a neighborhood of g₀.     

Almost hermitian manifolds and Osserman condition

Let(M, g, J) be an almost Hermitian manifold. In this paper we study holomorphically nonnegatively(δ,Δ)-pinched almost Hermitian manifolds. In [3] it was shown that for such Kahler manifolds a plane with maximal sectional curvature has to be a holomorphic plane(J-invariant). Here we generalize this result to arbitrary almost Hermitian manifolds with respect to the holomorphic curvature tensorH R and toRK-manifolds of a constant type λ(p). In the proof some estimates of the sectional curvature are established. The results obtained are used to characterize almost Hermitian manifolds of constant holomorphic sectional curvature (with respect to holomorphic and Riemannian curvature tensor) in terms of the eigenvalues of the Jacobi-type operators, i.e. to establish partial cases of the Osserman conjecture. Some examples are studied. The first author is partially supported by SFS, Project #04M03.     

On Hermitian curvature flow on almost complex manifolds

In the present paper we generalize the Hermitian curvature flow introduced and studied in Streets and Tian (2011) [6] to the almost complex case.     

The distribution of the root degree of a random permutation

Given a permutation ω of {1, …, n}, let R(ω) be the root degree of ω, i.e. the smallest (prime) integer r such that there is a permutation σ with ω = σ ^r . We show that, for ω chosen uniformly at random, R(ω) = (lnlnn − 3lnlnln n + O ^p (1))⁻¹ lnn, and find the limiting distribution of the remainder term. Research supported in part by NSF grants CCR-0225610, DMS-0505550 and ARO grant W911NF-06-1-0076. Research supported by NSF grant DMS-0406024.     

Clifford structures on Riemannian manifolds

We introduce the notion of even Clifford structures on Riemannian manifolds, which for rank r=2 and r=3 reduce to almost Hermitian and quaternion-Hermitian structures respectively. We give the complete classification of manifolds carrying parallel rank r even Clifford structures: Kähler, quaternion-Kähler and Riemannian products of quaternion-Kähler manifolds for r=2,3 and 4 respectively, several classes of 8-dimensional manifolds (for 5?r?8), families of real, complex and quaternionic Grassmannians (for r=8,6 and 5 respectively), and Rosenfeld?s elliptic projective planes OP², (C⊗O)P², (H⊗O)P² and (O⊗O)P², which are symmetric spaces associated to the exceptional simple Lie groups F₄, E₆, E₇ and E₈ (for r=9,10,12 and 16 respectively). As an application, we classify all Riemannian manifolds whose metric is bundle-like along the curvature constancy distribution, generalizing well-known results in Sasakian and 3-Sasakian geometry.     

Conformal paracontact curvature and the local flatness theorem

A curvature-type tensor invariant called para contact (pc) conformal curvature is defined on a paracontact manifold. It is shown that a paracontact manifold is locally paracontact conformal to the hyperbolic Heisenberg group or to a hyperquadric of neutral signature iff the pc conformal curvature vanishes. In the three dimensional case the corresponding result is achieved through employing a certain symmetric (0,2) tensor. The well known result of Cartan–Chern–Moser giving necessary and sufficient condition a CR-structure to be CR equivalent to a hyperquadric in \mathbbCⁿ⁺¹{\mathbb{C}^{n+1}} is presented in-line with the paracontact case. An explicit formula for the regular part of a solution to the sub-ultrahyperbolic Yamabe equation on the hyperbolic Heisenberg group is shown.