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1.
We give a classification of 3—dimensional conformally flat contact metric manifolds satisfying: =0(=L
g) orR(Y, Z)=k[(Z)Y–(Y)Z]+[(Z)hY]–(Y)hZ] wherek and are functions. It is proved that they are flat (the non-Sasakian case) or of constant curvature 1 (the Sasakian case). 相似文献
2.
Domenico Perrone 《Journal of Geometry》2000,69(1-2):180-191
Blair [5] has introduced special directions on a contact metric 3-manifolds with negative sectional curvature for plane sections containing the characteristic vector field and, when is Anosov, compared such directions with the Anosov directions. In this paper we introduce the notion of Anosov-like special directions on a contact metric 3-manifold. Such directions exist, on contact metric manifolds with negative -Ricci curvature, if and only if the torsion is -parallel, namely (1.1) is satisfied. If a contact metric 3-manifold M admits Anosov-like special directions, and is -parallel, where is the Berger-Ebin operator, then is Anosov and the universal covering of M is the Lie group
(2,R). We note that the notion of Anosov-like special directions is related to that of conformally Anosow flow introduced in [9] and [14] (see [6]).Supported by funds of the M.U.R.S.T. and of the University of Lecce. 1991. 相似文献
3.
We study the geometric properties of the base manifold for the unit tangent bundle satisfying the η-Einstein condition with
the canonical contact metric structure. One of the main theorems is that the unit tangent bundle of 4-dimensional Einstein
manifold, equipped with the canonical contact metric structure, is η-Einstein manifold if and only if the base manifold is
the space of constant sectional curvature 1 or 2.
Authors’ addresses: Y. D. Chai, S. H. Chun, J. H. Park, Department of Mathematics, Sungkyunkwan University, Suwon 440-746,
Korea; K. Sekigawa, Department of Mathematics, Faculty of Science, Niigata University, Niigata, 950-2181, Japan 相似文献
4.
Domenico Perrone 《Journal of Geometry》2005,83(1-2):164-174
We prove that on a compact (non Sasakian) contact metric 3-manifold with critical metric for the Chern-Hamilton functional,
the characteristic vector field ξ is conformally Anosov and there exists a smooth curve in the contact distribution of conformally
Anosov flows. As a consequence, we show that negativity of the ξ-sectional curvature is not a necessary condition for conformal
Anosovicity of ξ (this completes a result of [4]). Moreover, we study contact metric 3-manifolds with constant ξ-sectional
curvature and, in particular, correct a result of [13]. 相似文献
5.
Ricci solitons were introduced by R. Hamilton as natural generalizations of Einstein metrics. A Ricci soliton on a smooth
manifold M is a triple (g0,ξ, λ), where g0 is a complete Riemannian metric, ξ a vector field, and λ a constant such that the Ricci tensor Ric0 of the metric g0 satisfies the equation ò2 Ric0 = Lξg0 + 2λgo. The following statement is one of the main results of the paper. Let (g0,ξ, λ) be a Ricci soliton such that M,g0 is a complete noncompact oriented Riemannian manifold, $
\int\limits_M {\left\| \xi \right\|dv < \infty }
$
\int\limits_M {\left\| \xi \right\|dv < \infty }
, and the scalar curvature s0 of g0 has a constant sign on M, then (M, g0) is an Einstein manifold 相似文献
6.
We consider three-dimensional unimodular Lie groups equipped with a Lorentzian metric and we determine, for all of them, their
sets of homogeneous geodesics through a point.
Dedicated to the memory of Professor Aldo Cossu
Authors supported by funds of M.U.R.S.T., G.N.S.A.G.A. and the University of Lecce. 相似文献
7.
R. Chavosh Khatamy 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2010,45(1):60-65
The paper considers the associated bundle ξ = (G × KG/K, ρ
ξ
, G/K, G/K) and the tangent bundle τ
G/K
= (T
G/K
, π
G/K
, G/K, R
m
), and gives special examples of odd dimensional solvable Lie groups equipped with left invariant Riemannian metric. Some
conditions about existence of homogeneous geodesic vectors on the fiber space of ξ and τ
G/K
are proved. 相似文献
8.
We construct examples of symplectic half-flat manifolds on compact quotients of solvable Lie groups. We prove that the Calabi-Yau
structures are not rigid in the class of symplectic half-flat structures. Moreover, we provide an example of a compact 6-dimensional
symplectic half-flat manifold whose real part of the complex volume form is d-exact. Finally we discuss the 4-dimensional case.
This work was supported by the Projects M.I.U.R. “Geometric Properties of Real and Complex Manifolds”, “Riemannian Metrics
and Differentiable Manifolds” and by G.N.S.A.G.A. of I.N.d.A.M. 相似文献
9.
We study ball-homogeneity, curvature homogeneity, natural reductivity, conformal flatness and ϕ-symmetry for three-dimensional
contact metric manifolds. Several classification results are given.
Member of G.N.S.A.G.A.
Supported by funds of the M.U.R.S.T. 相似文献
10.
We consider a (2m + 3)-dimensional Riemannian manifold M(ξ r, ηr, g ) endowed with a vertical skew symmetric almost contact 3-structure. Such manifold is foliated by 3-dimensional submanifolds
of constant curvature tangent to the vertical distribution and the square of the length of the vertical structure vector
field is an isoparametric function. If, in addition, M(ξ r, ηr, g ) is endowed with an f -structure φ, M, turns out to be a framed f−CR-manifold. The fundamental 2-form Ω associated with φ is a presymplectic form. Locally, M is the Riemannian product
of two totally geodesic submanifolds, where
is a 2m-dimensional Kaehlerian submanifold and
is a 3-dimensional submanifold of constant curvature. If M is not compact, a class of local Hamiltonians of Ω is obtained. 相似文献
11.
Domenico Perrone 《Archiv der Mathematik》1997,68(4):347-352
Let CP
n
be the n-dimensional complex projective space with the Study-Fubini metric of constant holomorphic sectional curvature 4 and let M be a compact, orientable, n-dimensional totally real minimal submanifold of CP
n
. In this paper we prove the following results.
Supported by funds of the M.U.R.S.T. 相似文献
(a) | If M is 6-dimensional, conformally flat and has non negative Euler number and constant scalar curvature τ, 0<τ ≦ 70/3, then M is locally isometric to S 1,5 :=S 1 (sin θ cos θ) × S 5 (sin θ), tan θ = √6. |
(b) | If M is 4-dimensional, has parallel second fundamental form and scalar curvature τ ≧ 15/2, then M is locally isometric to S 1,3 :=S 1 (sin θ cos θ) × S 3 (sinθ), tan θ=2, or it is totally geodesic. |
12.
The PDE Ric(g) = λ · g for a Riemannian Einstein metric g on a smooth manifold M becomes an ODE if we require g to be invariant
under a Lie group G acting properly on M with principal orbits of codimension one. A singular orbit of the G-action gives
a singularity of this ODE. Generically, an equation with such type of singularity has no smooth solution at the singularity.
However, in our case, the very geometric nature of the equation makes it solvable. More precisely, we obtain a smooth G-invariant
Einstein metric (with any Einstein constant λ) in a tubular neighbourhood around a singular orbit Q ⊂ M for any prescribed
G-invariant metric gQ and second fundamental form LQ on Q, provided that the following technical condition is satisfied (which is very often the case): the representations of
the principal isotropy group on the tangent and the normal space of the singular orbit Q have no common sub-representations.
This Einstein metric is not uniquely determined by the initial data gQ and LQ; in fact, one may prescribe initial derivatives of higher degree, and examples show that this degree can be arbitrarily high.
The proof involves a blend of ODE techniques and representation theory of the principal and singular isotropy groups. 相似文献
13.
Abstract
Let A be a unital simple C*-algebra of real zero, stable rank one, with weakly unperforated K
0(
A) and unique normalized quasi-trace τ, and let X be a compact metric space. We show that two monomorphisms φ, ψ : C(X)→A are approximately unitarily equivalent if and only if φ and ψ induce the same element in KL(C(X), A) and the two lineal functionals τ∘φ and τ∘ψ are equal. We also show that, with an injectivity condition, an almost multiplicative
morphism from C(X) into A with vanishing KK-obstacle is close to a homomorphism.
Research partially supported by NSF Grants DMS 93-01082 (H.L) and DMS-9401515(G.G). This work was reported by the first named
author at West Coast Operator Algebras Seminar (Sept. 1995, Eugene, Oregon) 相似文献
14.
Dan Mangoubi 《Mathematische Annalen》2008,341(1):1-13
We consider Riemannian metrics compatible with the natural symplectic structure on T
2 × M, where T
2 is a symplectic 2-torus and M is a closed symplectic manifold. To each such metric we attach the corresponding Laplacian and consider its first positive
eigenvalue λ1. We show that λ1 can be made arbitrarily large by deforming the metric structure, keeping the symplectic structure fixed. The conjecture is
that the same is true for any symplectic manifold of dimension ≥ 4. We reduce the general conjecture to a purely symplectic
question. 相似文献
15.
We study the natural almost CR structure on the total space of a subbundle of hyperquadrics of the tangent bundle T(M) over a semi-Riemannian manifold (M, g) and show that if the Reeb vector ξ of an almost contact Riemannian manifold is a CR map then the natural almost CR structure on M is strictly pseudoconvex and a posteriori ξ is pseudohermitian. If in addition ξ is geodesic then it is a harmonic vector field. As an other application, we study pseudoharmonic vector fields on a compact strictly pseudoconvex CR manifold M, i.e. unit (with respect to the Webster metric associated with a fixed contact form on M) vector fields X ε H(M) whose horizontal lift X↑ to the canonical circle bundle S1 → C(M) → M is a critical point of the Dirichlet energy functional associated to the Fefferman metric (a Lorentz metric on C(M)). We show that the Euler–Lagrange equations satisfied by X↑ project on a nonlinear system of subelliptic PDEs on M.
Mathematics Subject Classifications (2000): 53C50, 53C25, 32V20 相似文献
16.
Gabriel Larotonda 《Integral Equations and Operator Theory》2006,54(4):511-523
The Hilbert manifold ∑ ∞ consisting of positive invertible (unitized) Hilbert-Schmidt operators has a rich structure and geometry. The geometry of
unitary orbits Ω⊂∑ ∞ is studied from the topological and metric viewpoints: we seek for conditions that ensure the existence of a smooth local
structure for the set Ω, and we study the convexity of this set for the geodesic structures that arise when we give ∑ ∞ different Riemannian metrics. 相似文献
17.
I. G. Nikolaev 《Commentarii Mathematici Helvetici》1995,70(1):210-234
Schur's theorem states that an isotropic Riemannian manifold of dimension greater than two has constant curvature. It is natural
to guess that compact almost isotropic Riemannian manifolds of dimension greater than two are close to spaces of almost constant
curvature. We take the curvature anisotropy as the discrepancy of the sectional curvatures at a point. The main result of
this paper is that Riemannian manifolds in Cheeger's class ℜ(n,d,V,A) withL
1-small integral anisotropy haveL
p-small change of the sectional curvature over the manifold. We also estimate the deviation of the metric tensor from that
of constant curvature in theW
p
2
-norm, and prove that compact almost isotropic spaces inherit the differential structure of a space form. These stability
results are based on the generalization of Schur' theorem to metric spaces. 相似文献
18.
We construct a three-parameter family of contact metric structures on the unit tangent sphere bundle T
1
M of a Riemannian manifold M and we study some of their special properties related to the Levi-Civita connection. More precisely, we give the necessary
and sufficient conditions for a constructed contact metric structure to be K-contact, Sasakian, to satisfy some variational conditions or to define a strongly pseudo-convex CR-structure. The obtained
results generalize classical theorems on the standard contact metric structure of T
1
M.
Author supported by funds of the University of Lecce. 相似文献
19.
Domenico Perrone 《Mathematische Zeitschrift》2009,263(1):125-147
It is well known that a Hopf vector field on the unit sphere S
2n+1 is the Reeb vector field of a natural Sasakian structure on S
2n+1. A contact metric manifold whose Reeb vector field ξ is a harmonic vector field is called an H-contact manifold. Sasakian and K-contact manifolds, generalized (k, μ)-spaces and contact metric three-manifolds with ξ strongly normal, are H-contact manifolds. In this paper we study, in dimension three, the stability with respect to the energy of the Reeb vector
field ξ for such special classes of H-contact manifolds (and with respect to the volume when ξ is also minimal) in terms of Webster scalar curvature. Finally, we extend for the Reeb vector field of a compact K-contact (2n+1)-manifold the obtained results for the Hopf vector fields to minimize the energy functional with mean curvature correction.
Supported by funds of the University of Lecce and M.I.U.R.(PRIN). 相似文献
20.
Gideon Maschler 《Differential Geometry and its Applications》2011,29(1):85-92
A Riemannian metric g with Ricci curvature r is called nontrivial quasi-Einstein, in a sense given by Case, Shu and Wei, if it satisfies (−a/f)∇df+r=λg, for a smooth nonconstant function f and constants λ and a>0. If a is a positive integer, it was noted by Besse that such a metric appears as the base metric for certain warped Einstein metrics. This equation also appears in the study of smooth metric measure spaces. We provide a local classification and an explicit construction of Kähler metrics conformal to nontrivial quasi-Einstein metrics, subject to the following conditions: local Kähler irreducibility, the conformal factor giving rise to a Killing potential, and the quasi-Einstein function f being a function of the Killing potential. Additionally, the classification holds in real dimension at least six. The metric, along with the Killing potential, form an SKR pair, a notion defined by Derdzinski and Maschler. It implies that the manifold is biholomorphic to an open set in the total space of a CP1 bundle whose base manifold admits a Kähler-Einstein metric. If the manifold is additionally compact, it is a total space of such a bundle or complex projective space. Additionally, a result of Case, Shu and Wei on the Kähler reducibility of nontrivial Kähler quasi-Einstein metrics is reproduced in dimension at least six in a more explicit form. 相似文献