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1.
Consider a compact Riemannian manifold (M, g) with metric g and dimension n ≥ 3. The Schouten tensor A
g
associated with g is a symmetric (0, 2)-tensor field describing the non-conformally-invariant part of the curvature tensor
of g. In this paper, we consider the elementary symmetric functions {σ
k
(A
g
), 1 ≤ k ≤ n} of the eigenvalues of A
g
with respect to g; we call σ
k
(A
g
) the k-th Schouten curvature function. We give an isometric classification for compact locally conformally flat manifolds which satisfy the conditions: A
g
is semi-positive definite and σ
k
(A
g
) is a nonzero constant for some k ∈ {2, ... , n}. If k = 2, we obtain a classification result under the weaker conditions that σ2(A
g
) is a non-negative constant and (M
n
, g) has nonnegative Ricci curvature. The corresponding result for the case k = 1 is well known. We also give an isometric classification for complete locally conformally flat manifolds with constant
scalar curvature and non-negative Ricci curvature.
Udo Simon: Partially supported by Chinese-German cooperation projects, DFG PI 158/4-4 and PI 158/4-5, and NSFC. 相似文献
2.
Thomas Church 《Inventiones Mathematicae》2012,188(2):465-504
Let C
n
(M) be the configuration space of n distinct ordered points in M. We prove that if M is any connected orientable manifold (closed or open), the homology groups H
i
(C
n
(M);ℚ) are representation stable in the sense of Church and Farb (). Applying this to the trivial representation, we obtain as a corollary that the unordered configuration space B
n
(M) satisfies classical homological stability: for each i, H
i
(B
n
(M);ℚ)≈H
i
(B
n+1(M);ℚ) for n>i. This improves on results of McDuff, Segal, and others for open manifolds. Applied to closed manifolds, this provides natural
examples where rational homological stability holds even though integral homological stability fails. 相似文献
3.
Wei Wang 《Annali di Matematica Pura ed Applicata》2007,186(2):359-380
By constructing normal coordinates on a quaternionic contact manifold M, we can osculate the quaternionic contact structure at each point by the standard quaternionic contact structure on the quaternionic
Heisenberg group. By using this property, we can do harmonic analysis on general quaternionic contact manifolds, and solve
the quaternionic contact Yamabe problem on M if its Yamabe invariant satisfies λ(M) < λ(ℍ
n
).
Mathematics Subject Classification (2000) 53C17, 53D10, 35J70 相似文献
4.
Let denote the universal covering space of a compact Riemannian manifold, M
n
, with sectional curvature, −1≤K
Mn
≤0. We show that a collection of deck transformations of , satisfying certain (metric dependent) conditions, determines an open dense subset of M
n
, at every point of which, there exists a local isometric splitting with nontrivial flat factor. Such a collection, which
we call an abelian structure, also gives rise to an essentially canonical Cr-structure in the sense of Buyalo, i.e an atlas for an injective F-structure, for which additional conditions hold. It follows
in particular that the minimal volume of M
n
vanishes. We show that an abelian structure exists if the injectivity radius at all points of M
n
is less than ε(n)>0. This yields a conjecture of Buyalo as well as a strengthened version of the conclusion of Gromov’s “gap conjecture” in
our special situation. In addition, we observe that abelian structures on nonpositively curved manifolds have certain stability
properties under suitably controlled changes of metric.
Oblatum 26-III-1999 & 14-IX-2000?Published online: 8 December 2000 相似文献
5.
Nathan Smale 《Inventiones Mathematicae》1999,135(1):145-183
For n≥7, it is shown how to construct examples of smooth, compact Riemannian manifolds (N
n
+1,g), with non-trivial n dimensional integer homology, such that for some Γ∈H
n
(N,Z), the hypersurface (n-current) M, which minimizes area among all hypersurfaces representing Γ, has singularities. The singular set of M consists of two isolated points, and the tangent cone at these points can be prescribed as any strictly stable, strictly
minimizing, regular cone. To my knowledge these are the first examples of codimension one homological minimizers with singularities.
Oblatum: 3-I-1997 & 13-II-1998 / Published online: 18 September 1998 相似文献
6.
Michal Sadowski 《Central European Journal of Mathematics》2004,2(2):332-338
Let E
Aff(Γ,G, m) be the set of affine equivalence classes of m-dimensional complete flat manifolds with a fixed fundamental group Γ and a fixed holonomy group G. Let n be the dimension of a closed flat manifold whose fundamental group is isomorphic to Γ. We describe E
Aff(Γ,G, m) in terms of equivalence classes of pairs (ε, ρ), consisting of epimorphisms of Γ onto G and representations of G in ℝ
m-n
. As an application we give some estimates of card E
Aff(Γ,G, m). 相似文献
7.
《Selecta Mathematica, New Series》2005,11(3-4):399-451
We verify the conjecture formulated in [36] for suspension singularities of type g(x, y, z)=f(x, y)+zn, where f is an irreducible plane curve singularity. More precisely, we prove that the modified Seiberg–Witten invariant of the link
M of g, associated with the canonical spinc structure, equals −σ(F)/8, where σ(F) is the signature of the Milnor fiber of g. In order to do this, we prove general splicing formulae for the Casson–Walker invariant and for the sign-refined Reidemeister–Turaev
torsion. These provide results for some cyclic covers as well. As a by-product, we compute all the relevant invariants of
M in terms of the Newton pairs of f and the integer n. 相似文献
8.
Marc Arnaudon 《Probability Theory and Related Fields》1997,108(2):219-257
Summary. We prove that the derivative of a differentiable family X
t
(a) of continuous martingales in a manifold M is a martingale in the tangent space for the complete lift of the connection in M, provided that the derivative is bicontinuous in t and a. We consider a filtered probability space (Ω,(ℱ
t
)0≤
t
≤1, ℙ) such that all the real martingales have a continuous version, and a manifold M endowed with an analytic connection and such that the complexification of M has strong convex geometry. We prove that, given an analytic family a↦L(a) of random variable with values in M and such that L(0)≡x
0∈M, there exists an analytic family a↦X(a) of continuous martingales such that X
1(a)=L(a). For this, we investigate the convexity of the tangent spaces T
(
n
)
M, and we prove that any continuous martingale in any manifold can be uniformly approximated by a discrete martingale up to
a stopping time T such that ℙ(T<1) is arbitrarily small. We use this construction of families of martingales in complex analytic manifolds to prove that
every ℱ1-measurable random variable with values in a compact convex set V with convex geometry in a manifold with a C
1 connection is reachable by a V-valued martingale.
Received: 14 March 1996/In revised form: 12 November 1996 相似文献
9.
We construct irreducible pseudo-Riemannian manifolds (M, g) of arbitrary signature (p, q) with the same curvature tensor as a pseudo-Riemannian symmetric space which is a direct product of a two-dimensional Riemannian
space form M
2(c) and a pseudo-Euclidean space with the signature (p, q − 2), or (p − 2, q), respectively. 相似文献
10.
Baruch Solel 《Israel Journal of Mathematics》1988,62(1):63-89
LetM be a σ-finite von Neumann algebra andα be an action ofR onM. LetH
∞(α) be the associated analytic subalgebra; i.e.H
∞(α)={X ∈M: sp∞(X) [0, ∞]}. We prove that every σ-weakly closed subalgebra ofM that containsH
∞(α) isH
∞(γ) for some actionγ ofR onM. Also we show that (assumingZ(M)∩M
α = Ci)H
∞(α) is a maximal σ-weakly closed subalgebra ofM if and only if eitherH
∞(α)={A ∈M: (I−F)xF=0} for some projectionF ∈M, or sp(α)=Γ(α). 相似文献
11.
Jimmy Petean 《Geometriae Dedicata》2009,143(1):37-48
We study isoperimetric regions on Riemannian manifolds of the form (M
n
× (0, π), sin2(t)g + dt
2) where g is a metric of positive Ricci curvature ≥ n − 1. When g is an Einstein metric we use this to compute the Yamabe constant of
(M ×\mathbbR, g+ dt2 ){(M \times \mathbb{R}, g+ dt^2 )} and so to obtain lower bounds for the Yamabe invariant of M × S
1. 相似文献
12.
Let σ(n) be the minimum number of ideal hyperbolic tetrahedra necessary to construct a finite volumen-cusped hyperbolic 3-manifold, orientable or not. Let σor(n) be the corresponding number when we restrict ourselves to orientable manifolds. The correct values of σ(n) and σor(n) and the corresponding manifolds are given forn=1,2,3,4 and 5. We then show that 2n−1≤σ(n)≤σor(n)≤4n−4 forn≥5 and that σor(n)≥2n for alln.
Both authors were supported by NSF Grants DMS-8711495, DMS-8802266 and Williams College Research Funds. 相似文献
13.
For κ ⩾ 0 and r0 > 0 let ℳ(n, κ, r0) be the set of all connected, compact n-dimensional Riemannian manifolds (Mn, g) with Ricci (M, g) ⩾ −(n−1) κ g and Inj (M) ⩾ r0. We study the relation between the kth eigenvalue λk(M) of the Laplacian associated to (Mn,g), Δ = −div(grad), and the kth eigenvalue λk(X) of a combinatorial Laplacian associated to a discretization X of M. We show that there exist constants c, C > 0 (depending only on n, κ and r0) such that for all M ∈ ℳ(n, κ, r0) and X a discretization of
for all k < |X|. Then, we obtain the same kind of result for two compact manifolds M and N ∈ ℳ(n, κ, r0) such that the Gromov–Hausdorff distance between M and N is smaller than some η > 0. We show that there exist constants c, C > 0 depending on η, n, κ and r0 such that
for all
.
Mathematics Subject Classification (2000): 58J50, 53C20
Supported by Swiss National Science Foundation, grant No. 20-101 469 相似文献
14.
Michael T. Anderson 《Selecta Mathematica, New Series》2010,16(3):343-375
We investigate the validity of the isometry extension property for (Riemannian) Einstein metrics on compact manifolds M with boundary ∂M. Given a metric γ on ∂M, this is the issue of whether any Killing field X of (∂M, γ) extends to a Killing field of any Einstein metric (M, g) bounding (∂M, γ). Under a mild condition on the fundamental group, this is proved to be the case at least when X preserves the mean curvature of ∂M in (M, g). 相似文献
15.
György Gát 《数学学报(英文版)》2007,23(12):2269-2294
A highly celebrated problem in dyadic harmonic analysis is the pointwise convergence of
the Fejér (or (C, 1)) means of functions on unbounded Vilenkin groups. There are several papers of
the author of this paper concerning this. That is, we know the a.e. convergence σ
n
f → f (n → ∞)
for functions f ∈ L
p
, where p > 1 (Journal of Approximation Theory, 101(1), 1–36, (1999)) and also
the a.e. convergence σM
n
f → f (n → ∞) for functions f ∈ L
1 (Journal of Approximation Theory,
124(1), 25–43, (2003)). The aim of this paper is to prove the a.e. relation lim
n
→ σ
n
f = f for each
integrable function f on any rarely unbounded Vilenkin group. The concept of the rarely unbounded
Vilenkin group is discussed in the paper. Basically, it means that the generating sequence m may be
an unbounded one, but its "big elements" are not "too dense".
Research supported by the Hungarian National Foundation for Scientific Research (OTKA), Grant No. M
36511/2001 and T 048780 相似文献
16.
Amalendu Ghosh Ramesh Sharma Jong Taek Cho 《Annals of Global Analysis and Geometry》2008,34(3):287-299
We show that a non-Sasakian contact metric manifold with η-parallel torsion tensor and sectional curvatures of plane sections containing the Reeb vector field different from 1 at some
point, is a (k, μ)-contact manifold. In particular for the standard contact metric structure of the tangent sphere bundle the torsion tensor
is η-parallel if and only if M is of constant curvature, in which case its associated pseudo-Hermitian structure is CR- integrable. Next we show that if
the metric of a non-Sasakian (k, μ)-contact manifold (M, g) is a gradient Ricci soliton, then (M, g) is locally flat in dimension 3, and locally isometric to E
n+1 × S
n
(4) in higher dimensions.
相似文献
17.
An equivelar polyhedral 2-manifold in the classM
p,q is one embedded inE
3 in which every face is a convexp-gon and every vertex isq-valent. Various constructions for equivelar manifolds are described, and it is shown that, in certain classesM
p,q, there is a manifold of given genusg≧2 for all but finitely manyg. 相似文献
18.
SoitM(Ω, η, ξ,g) une variété à (2m+1)-dimensions presque cosymplectique (i. e. Ω∈Λ2
M est de rang 2m et Ω
m
Λη≠0). On définitM comme étant une variété semi-cosymplectique si en termes ded
ω-cohomologie la paire (Ω, η) satisfait àdη=0,d
−cη Ω=Ψ∈Λ3
M,c=constant. Dans ce cas le champ vectoriel de structure ξ=b
−1(η) est un champ conforme horizontal et siM est une forme-espace elle est nécessairement du type hyperbolique. Différentes propriétés de cette structure sont étudiés
et le cas oùM est une variété para Sasakienne dans le sens large est discuté. 相似文献
19.
In this paper, we establish some sharp Sobolev trace inequalities on n-dimensional, compact Riemannian manifolds with smooth boundaries. More specifically, let q = 2(n - 1)/(n - 2), 1/S = inf {∫ |∇u|2 : ∇u ∈ L2(R+n), ∫ |u|q = 1}. We establish for any Riemannian manifold with a smooth boundary, denoted as (M, g), that there exists some constant A = A(M, g) > 0, (∫dM|u|q dsg)2/q < or = to S ∫M |∇gu|2 dvg + A ∫dMu2 dsg, for all u ∈ H1 (M). The inequality is sharp in the sense that the inequality is false when S is replaced by any smaller number. © 1997 John Wiley & Sons, Inc. 相似文献
20.
Isabella Novik 《Israel Journal of Mathematics》1998,108(1):45-82
In this paper we prove the Upper Bound Conjecture (UBC) for some classes of (simplicial) homology manifolds: we show that
the UBC holds for all odd-dimensional homology manifolds and for all 2k-dimensional homology manifolds Δ such that β
k
(Δ)⩽Σ{β
i
(Δ):i ≠k-2,k,k+2 and 1 ⩽i⩽2k-1}, where β
i
(Δ) are reduced Betti numbers of Δ. (This condition is satisfied by 2k-dimensional homology manifolds with Euler characteristic χ≤2 whenk is even or χ≥2 whenk is odd, and for those having vanishing middle homology.)
We prove an analog of the UBC for all other even-dimensional homology manifolds.
Kuhnel conjectured that for every 2k-dimensional combinatorial manifold withn vertices,
. We prove this conjecture for all 2k-dimensional homology manifolds withn vertices, wheren≥4k+3 orn≤3k+3. We also obtain upper bounds on the (weighted) sum of the Betti numbers of odd-dimensional homology manifolds. 相似文献