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1.
In this paper, we study ruled Weingarten surfaces M : x (s, t) = α(s) + tβ (s) in Minkowski 3-space on which there is a nontrivial functional relation between a pair of elements of the set {K, KII, H, HII}, where K is the Gaussian curvature, KII is the second Gaussian curvature, H is the mean curvature, and HII is the second mean curvature. We also study ruled linear Weingarten surfaces in Minkowski 3-space such that the linear combination
aKII + bH + cHII + dK is constant along each ruling for some constants a, b, c, d with a2 + b2 + c2 ≠ 0. 相似文献
2.
Krishan L. Duggal 《Central European Journal of Mathematics》2012,10(5):1789-1800
This paper deals with a family of lightlike (null) hypersurfaces (H u ) of a Lorentzian manifold M such that each null normal vector ℓ of H u is not entirely in H u , but, is defined in some open subset of M around H u . Although the family (H u ) is not unique, we show, subject to some reasonable condition(s), that the involved induced objects are independent of the choice of (H u ) once evaluated at u = constant. We use (n+1)-splitting Lorentzian manifold to obtain a normalization of ℓ and a well-defined projector onto H, needed for Gauss, Weingarten, Gauss-Codazzi equations and calculate induced metrics on proper totally umbilical and totally geodesic H u . Finally, we establish a link between the geometry and physics of lightlike hypersurfaces and a variety of black hole horizons. 相似文献
3.
Kensuke Onda 《Geometriae Dedicata》2010,147(1):313-322
The three-dimensional Heisenberg group H
3 has three left-invariant Lorentzian metrics g
1, g
2, and g
3 as in Rahmani (J. Geom. Phys. 9(3), 295–302 (1992)). They are not isometric to each other. In this paper, we characterize
the left-invariant Lorentzian metric g
1 as a Lorentz Ricci Soliton. This Ricci Soliton g
1 is a shrinking non-gradient Ricci Soliton. We also prove that the group E(2) of rigid motions of Euclidean 2-space and the group E(1, 1) of rigid motions of Minkowski 2-space have Lorentz Ricci Solitons. 相似文献
4.
Lorenz J. Schwachhöfer 《Geometriae Dedicata》1996,62(2):193-208
In Proc. Symp. Pure Math.
53 (1991), 33–88, Bryant gave examples of torsion free connections on four-manifolds whose holonomy is exotic, i.e. is not contained on Berger's classical list of irreducible holonomy representations. The holonomy in Bryant's examples is the irreducible four-dimensional representation of S1(2, #x211D;) (G1(2, #x211D;) resp.) and these connections are called H
3-connections (G
3-connections resp.).In this paper, we give a complete classification of homogeneous G
3-connections. The moduli space of these connections is four-dimensional, and the generic homogeneous G
3-connection is shown to be locally equivalent to a left-invariant connection on U(2). Thus, we prove the existence of compact manifolds with G
3-connections. This contrasts a result in by Schwachhöfer (Trans. Amer. Math. Soc.
345 (1994), 293–321) which states that there are no compact manifolds with an H
3-connection. 相似文献
5.
Miguel Angel Javaloyes Levi Lopes de Lima Paolo Piccione 《Mathematische Zeitschrift》2008,260(2):277-303
Following the lines of Bott in (Commun Pure Appl Math 9:171–206, 1956), we study the Morse index of the iterates of a closed
geodesic in stationary Lorentzian manifolds, or, more generally, of a closed Lorentzian geodesic that admits a timelike periodic
Jacobi field. Given one such closed geodesic γ, we prove the existence of a locally constant integer valued map Λγ on the unit circle with the property that the Morse index of the iterated γ
N
is equal, up to a correction term εγ∈{0,1}, to the sum of the values of Λγ at the N-th roots of unity. The discontinuities of Λγ occur at a finite number of points of the unit circle, that are special eigenvalues of the linearized Poincaré map of γ.
We discuss some applications of the theory. 相似文献
6.
In this paper, by defining Clifford algebra product in 3-dimensional Lorentz space, L
3, it is shown that even Clifford algebra of L
3 corresponds to split quaternion algebra. Then, by using Lorentzian matrix multiplication, pole point of planar displacement
in Lorentz plane L
2 is obtained. In addition, by defining degenerate Lorentz scalar product for L
3 and by using the components of pole points of Lorentz plane displacement in particular split hypercomplex numbers, it is
shown that the Lorentzian planar displacements can be represented as a special split quaternion which we call it Lorentzian
planar split quaternion.
相似文献
7.
A submanifold M
n
r
of Minkowski space
is said to be of restricted type if its shape operator with respect to the mean curvature vector is the restriction of a fixed linear transformation of
to the tangent space of M
n
r
at every point of M
n
r
. In this paper we completely classify hypersurfaces of restricted type in
. More precisely, we prove that a hypersurface of
is of restricted type if and only if it is either a minimal hypersurface, or an open part of one of the following hypersurfaces: S
k
×
, S
k
1
×
, H
k
×
, S
n
1
, H
n
, with 1kn–1, or an open part of a cylinder on a plane curve of restricted type.This work was done when the first and fourth authors were visiting Michigan State University.Aangesteld Navorser N.F.W.O., Belgium. 相似文献
8.
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. 相似文献
9.
ZhangJianfeng 《高校应用数学学报(英文版)》2005,20(2):183-196
Let M^n be a closed spacelike submanifold isometrically immersed in de Sitter space Sp^(n p)(c), Denote by R,H and S the normalized scalar curvature,the mean curvature and the square of the length of the second fundamental form of M^n ,respectively. Suppose R is constant and R≤c. The pinching problem on S is studied and a rigidity theorem for M^n immersed in Sp^(n p)(c) with parallel normalized mean curvature vector field is proved. When n≥3, the pinching constant is the best. Thus, the mistake of the paper “Space-like hypersurfaces in de Sitter space with constant scalar curvature”(see Manus Math, 1998,95 :499-505) is corrected. Moreover,the reduction of the codimension when M^n is a complete submanifold in Sp^(n p)(c) with parallel normalized mean curvature vector field is investigated. 相似文献
10.
We prove that the universal covering spaces of the generic submanifolds
of C
P
n
and
of C
H
n
are naturally reductive homogeneous spaces by determining explicitly tensor fields defining naturally reductive homogeneous structures on them. 相似文献
11.
Christian GroßRID=""ID=""Supported by a DFG grant. 《manuscripta mathematica》2000,103(3):339-350
Let denote the eigenspace decomposition of a twisted affine Kac–Moody algebra with respect to an involution , where is a twisted loop algebra, is the center and d is the scaling element of . We endow with the standard bilinear symmetrical form.Then with and carries a Lorentzian signature. Let denote the group that corresponds to , then the adjoint representation of on can be restricted to and this submanifold is isometrical to the Hilbert space E
ε, where is the decomposition of the twisted loop algebra with respect to the induced involutionρ0.We thus obtain an affine representation on E
ε and we show that this representation is polar, i. e., there exists a submanifold that intersects all orbits, and intersects
them orthogonally.
Received: 16 February 2000
RID="
ID="Supported by a DFG grant. 相似文献
12.
John A. Velling 《Journal of Geometric Analysis》1999,9(3):457-489
A set of conditions are given, each equivalent to the constancy of mean curvature of a surface in
H
3.It is shown that analogs of these equivalences exist for surfaces in
S
∞
2
,the bounding ideal sphere of
H
3,leading to a notion of constant mean curvature at infinity of
H
3.A parametrization of all complete constant mean curvature surfaces at infinity of
H
3
is given by holomorphic quadratic differentials on Ĉ,C, and
D. 相似文献
13.
Lorenzo Mazzieri 《Calculus of Variations and Partial Differential Equations》2009,34(4):453-473
In this paper we construct a family of new (topologically distinct) solutions to the Einstein constraint equations by performing
the generalized connected sum (or fiber sum) of two known compact m-dimensional constant mean curvature solutions (M
1, g
1, Π1) and (M
2, g
2,Π2) along a common isometrically embedded k-dimensional sub-manifold (K, g
K
). Away from the gluing locus the metric and the second fundamental form of the new solutions can be chosen as close as desired
to the ones of the original solutions. The proof is essentially based on the conformal method and the geometric construction
produces a polyneck between M
1 and M
2 whose metric is modeled fiber-wise (i. e. along the slices of the normal fiber bundle of K) around a Schwarzschild metric; for these reasons the codimension n : = m − k of K in M
1 and M
2 is required to be ≥ 3. In this sense our result is a generalization of the Isenberg–Mazzeo–Pollack gluing, which works
for connected sum at points and in dimension 3. The solutions we obtain for the Einstein constraint equations can be used
to produce new short time vacuum solutions of the Einstein system on a Lorentzian (m + 1)-dimensional manifold, as guaranteed by a well known result of Choquet-Bruhat. 相似文献
14.
Yun Tao Zhang 《Differential Geometry and its Applications》2011,29(6):730-736
Let Mn be a complete hypersurface in Sn+1(1) with constant mean curvature. Assume that Mn has n−1 principal curvatures with the same sign everywhere. We prove that if RicM≤C−(H), either S?S+(H) or RicM?0 or the fundamental group of Mn is infinite, then S is constant, S=S+(H) and Mn is isometric to a Clifford torus with . These rigidity theorems are still valid for compact hypersurface without constancy condition on the mean curvature. 相似文献
15.
Hojoo Lee 《Geometriae Dedicata》2011,151(1):373-386
As a generalization of the classical duality between minimal graphs in E
3 and maximal graphs in L
3, we construct the duality between graphs of constant mean curvature H in Bianchi-Cartan-Vranceanu space E
3(κ, τ) and spacelike graphs of constant mean curvature τ in Lorentzian Bianchi-Cartan-Vranceanu space L
3(κ, H). 相似文献
16.
Qun Chen 《manuscripta mathematica》1998,95(1):507-517
LetM, N be complete manifolds,u:M →N be a harmonic map with potentialH, namely, a critical point of the functionalE
H
(u)=∫
M
[e(u) − H(u)], wheree(u) is the energy density ofu. We will give a Liouville theorem foru with a class of potentialsH’s.
Research supported in part by NNSFC, SFECC and NSFCCNU. 相似文献
17.
18.
Uniqueness of complete spacelike hypersurfaces via their higher order mean curvatures in a conformally stationary spacetime
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Henrique Fernandes de Lima Marco Antonio Lázaro Velásquez 《Mathematische Nachrichten》2014,287(11-12):1223-1240
We study complete noncompact spacelike hypersurfaces immersed into conformally stationary spacetimes, that is, Lorentzian manifolds endowed with a timelike conformal vector field V. In this setting, by using as main analytical tool a suitable maximum principle for complete noncompact Riemannian manifolds, we establish new characterizations of totally umbilical hypersurfaces in terms of their higher order mean curvatures. For instance, supposing an appropriated restriction on the norm of the tangential component of the vector field V, we are able to show that such hypersurfaces must be totally umbilical provided that either some of their higher order mean curvatures are linearly related or one of them is constant. Applications to the so‐called generalized Robertson‐Walker spacetimes are given. In particular, we extend to the Lorentzian context a classical result due to Jellett 29 . 相似文献
19.
Daniel Allcock 《Inventiones Mathematicae》2000,140(2):283-301
We construct a natural sequence of finite-covolume reflection groups acting on the complex hyperbolic spaces ℂH
13, ℂH
9 and ℂH
5, and show that the 9-dimensional example coincides with the largest of the groups of Mostow [11]. Our reflection groups arise
as automorphism groups of certain Lorentzian lattices over the Eisenstein integers, and we obtain our largest example by using
the complex Leech lattice in a manner inspired by Conway [5]. We also construct finite-covolume reflection groups on the quaternionic
hyperbolic spaces ?H
7, ?H
5 and ?H
3, again using the Leech lattice, and apply results of Borcherds [4] to obtain automorphic forms for our groups.
Oblatum 25-III-1999 & 2-IX-1999?Published online: 21 February 2000 相似文献
20.
Starting from a 6-dimensional nilpotent Lie group N endowed with an invariant SU(3) structure, we construct a homogeneous conformally parallel G2-metric on an associated solvmanifold. We classify all half-flat SU(3) structures that endow the rank-one solvable extension of N with a conformally parallel G2 structure. By suitably deforming the SU(3) structures obtained, we are able to describe the corresponding non-homogeneous Ricci-flat metrics with holonomy contained
in G2. In the process we also find a new metric with exceptional holonomy.
Received: 20 September 相似文献