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1.
Spacelike hypersurfaces with constant scalar curvature 总被引:1,自引:0,他引:1
In this paper, we shall give an integral equality by applying the operator □ introduced by S.Y. Cheng and S.T. Yau [7] to
compact spacelike hypersurfaces which are immersed in de Sitter space S
n
+1
1(c) and have constant scalar curvature. By making use of this integral equality, we show that such a hypersurface with constant
scalar curvature n(n-1)r is isometric to a sphere if r << c.
Received: 18 December 1996 / Revised version: 26 November 1997 相似文献
2.
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
3.
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. |
4.
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
5.
Sharp estimates for the Ricci curvature of a submanifold M
n
of an arbitrary Riemannian manifold N
n+p
are established. It is shown that the equality in the lower estimate of the Ricci curvature of M
n
in a space form N
n+p
(c) is achieved only when M
n
is quasiumbilical with a flat normal bundle. In the case when the codimension p satisfies 1 ≤ p ≤ n − 3, the only submanifolds in N
n+p
(c) on which the Ricci curvature is minimal are the conformally flat ones with a flat normal bundle.
相似文献
6.
We control the number of critical points of a height function arising from the Nash isometric embedding of a compact Riemanniann-manifoldM. The Ln/2 curvature norm ∥R∥ and a similar scalar ∥R∥ are introduced and their integralR(M) andR(M) overM. We prove thatR(M) is bounded below by a constant depending only onn and the Betti numbers ofM. Thus a new sphere theorem is proved by eliminating allith Betti numbers fori = 1, .…n −1. The emphasis is that our sphere theorem imposes no restriction on the range of curvature.
Research partially supported by Grant-in-Aid for General Scientific Research, grant no. 07454018. 相似文献
7.
We prove that there exist (n − 1)-dimensional compact embedded rotational hypersurfaces with constant scalar curvature (n − 1)(n − 2)S (S > 1) of S
n
other than product of spheres for 4 ≤ n ≤ 6. As a result, we prove that Leite’s Assertion was incorrect.The project is supported by the grant No. 10531090 of NSFC. 相似文献
8.
In this paper, we are interested in extending the study of spherical curves in R
3 to the submanifolds in the Euclidean space R
n+p
. More precisely, we are interested in obtaining conditions under which an n-dimensional compact submanifold M of a Euclidean space R
n+p
lies on the hypersphere S
n+p−1(c) (standardly imbedded sphere in R
n+p
of constant curvature c). As a by-product we also get an estimate on the first nonzero eigenvalue of the Laplacian operator Δ of the submanifold
(cf. Theorem 3.5) as well as a characterization for an n-dimensional sphere S
n
(c) (cf. Theorem 4.1). 相似文献
9.
Chintamani M. N. Moriya B. K. Gao W. D. Paul P. Thangadurai R. 《Archiv der Mathematik》2012,98(2):133-142
Let G be a finite abelian group (written additively) of rank r with invariants n
1, n
2, . . . , n
r
, where n
r
is the exponent of G. In this paper, we prove an upper bound for the Davenport constant D(G) of G as follows; D(G) ≤ n
r
+ n
r-1 + (c(3) − 1)n
r-2 + (c(4) − 1) n
r-3 + · · · + (c(r) − 1)n
1 + 1, where c(i) is the Alon–Dubiner constant, which depends only on the rank of the group
\mathbb Znri{{\mathbb Z}_{n_r}^i}. Also, we shall give an application of Davenport’s constant to smooth numbers related to the Quadratic sieve. 相似文献
10.
Changyu Xia 《Monatshefte für Mathematik》2005,146(2):159-168
Let Sn(c) denote the n-dimensional Euclidean sphere of constant sectional curvature c and denote by CPn(c) the complex projective space of complex dimension n and of holomorphic sectional curvature c. In this paper, we obtain some characterizations of the manifolds S2(c) × S2(c′), S4(c) × S4(c′), CP2(c) × CP2(c′) by their spectrum. 相似文献
11.
In this paper, we classify complete spacelike hypersurfaces in the anti-de Sitter space (n?3) with constant scalar curvature and with two principal curvatures. Moreover, we prove that if Mn is a complete spacelike hypersurface with constant scalar curvature n(n−1)R and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is n−1, then R<(n−2)c/n. Additionally, we also obtain several rigidity theorems for such hypersurfaces. 相似文献
12.
Solving an old conjecture of Szele we show that the maximum number of directed Hamiltonian paths in a tournament onn vertices is at mostc · n
3/2
· n!/2
n–1, wherec is a positive constant independent ofn.Research supported in part by a U.S.A.-Israel BSF grant and by a Bergmann Memorial Grant. 相似文献
13.
In this paper we characterize the spacelike hyperplanes in the Lorentz–Minkowski space L
n
+1 as the only complete spacelike hypersurfaces with constant mean curvature which are bounded between two parallel spacelike
hyperplanes. In the same way, we prove that the only complete spacelike hypersurfaces with constant mean curvature in L
n
+1 which are bounded between two concentric hyperbolic spaces are the hyperbolic spaces. Finally, we obtain some a priori estimates
for the higher order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in
L
n
+1 which is bounded by a hyperbolic space. Our results will be an application of a maximum principle due to Omori and Yau, and
of a generalization of it.
Received: 5 July 1999 相似文献
14.
Marty Ross 《Journal of Geometric Analysis》1998,8(2):313-317
Let S ⊂ ℝn be a complete 2-dimensional areaminimizing mod 2 surface. Then S = x1 (M1) ∪ … ∪ xr (Mr) where each Mj is connected, xj: Mj → Vj is a classical minimal immersion into an affine subspace Vj of ℝn, and the subspaces V1,…, Vr are pairwise orthogonal. Here we prove that if Mj is orientable, then xj (Mj) is either aflat plane or, in suitable coordinates, a generalized complex hyperbola. 相似文献
15.
Given an integer q≥2, we say that a positive integer is a q-Niven number if it is divisible by the sum of its digits in base q. Given an arbitrary integer r∈[2,2q], we say that (n,n+1,…,n+r−1) is a q-Niven
r
-tuple if each number n+i, for i=0,1,…,r−1, is a q-Niven number. We show that there exists a positive constant c=c(q,r) such that the number of q-Niven r-tuples whose leading component is <x is asymptotic to cx/(log x)
r
as x→∞.
Research of J.M. De Koninck supported in part by a grant from NSERC.
Research of I. Kátai supported by the Applied Number Theory Research Group of the Hungarian Academy of Science and by a grant
from OTKA. 相似文献
16.
We establish integral formulas of Minkowski's type for compact spacelike hypersurfaces in de sitter spaceS
1
n+1
(1) and give their applications to the case of constantr-th mean curvature (r=1,2,…,n−1). Whenr=1 we recover Montiel's result.
Li Haizhong is supported by NNSFC No.19701017 and Basic Science Research Foundation of Tsinghua University and Chen Weihua
is supported by NNSFC No. 19571005 相似文献
17.
Andrew Suk 《Order》2010,27(1):63-68
Let r(n) denote the largest integer such that every family C\mathcal{C} of n pairwise disjoint segments in the plane in general position has r(n) members whose order type can be represented by points. Pach and Tóth gave a construction that shows r(n) < n
log8/log9 (Pach and Tóth 2009). They also stated that one can apply the Erdős–Szekeres theorem for convex sets in Pach and Tóth (Discrete Comput Geom 19:437–445,
1998) to obtain r(n) > log16
n. In this note, we will show that r(n) > cn
1/4 for some absolute constant c. 相似文献
18.
Let Гr,n—r denote the infimum of all number Г > 0 such that for any real indefinite quadratic form inn variables of type (r, n—r), determinantD ≠ 0 and real numbers c1; c2,…, cn, there exist integersx
1,x2,…,xn satisfying 0 < Q(x1+c1,x2 + c2,…,xn + cn) ≤(Г|Z > |)1/n. All the values of Гr,n—r are known except for г1,4. Earlier it was shown that 8 ≤Г1,4 ≤16. Here we improve the upper bound to get Г1,4 < 12. 相似文献
19.
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
20.
Erik Talvila 《Czechoslovak Mathematical Journal》2012,62(1):77-104
Let B
c
denote the real-valued functions continuous on the extended real line and vanishing at −∞. Let B
r
denote the functions that are left continuous, have a right limit at each point and vanish at −∞. Define A
c
n
to be the space of tempered distributions that are the nth distributional derivative of a unique function in B
c
. Similarly with A
r
n
from B
r
. A type of integral is defined on distributions in A
c
n
and A
r
n
. The multipliers are iterated integrals of functions of bounded variation. For each n ∈ ℕ, the spaces A
c
n
and A
r
n
are Banach spaces, Banach lattices and Banach algebras isometrically isomorphic to B
c
and B
r
, respectively. Under the ordering in this lattice, if a distribution is integrable then its absolute value is integrable.
The dual space is isometrically isomorphic to the functions of bounded variation. The space A
c
1 is the completion of the L
1 functions in the Alexiewicz norm. The space A
r
1 contains all finite signed Borel measures. Many of the usual properties of integrals hold: H?lder inequality, second mean
value theorem, continuity in norm, linear change of variables, a convergence theorem. 相似文献