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1.
Jianguo Cao 《Frontiers of Mathematics in China》2008,3(4):475-494
In this paper, we study certain compact 4-manifolds with non-negative sectional curvature K. If s is the scalar curvature and W. is the self-dual part of Weyl tensor, then it will be shown that there is no metric g on S
• × S
• with both (i) K > 0 and (ii) ÷ s − W
• ⩾ 0. We also investigate other aspects of 4-manifolds with non-negative sectional curvature. One of our results implies a
theorem of Hamilton: “If a simply-connected, closed 4-manifold M• admits a metric g of non-negative curvature operator, then M• is one of S•, ℂP• and S•×S•”. Our method is different from Hamilton’s and is much simpler. A new version of the second variational formula for minimal
surfaces in 4-manifolds is proved.
相似文献
2.
The total curvature of a compact C∞-immersed surface in Euclidean 3-space
3 can be interpreted as the average number of critical points for a linear ‘height’ function. The Morse inequalities provide
an intrinsic topological lower bound for the total curvature and ‘tight’ surfaces, which realize equality, have been an active
topic of research. The objective of this paper is to describe the natural notion of total curvature for C∞-singular surfaces which fail to immerse on C∞-embedded closed curves, but which have a
C∞-globally defined unit normal (e.g. caustics, or critical images for mappings of 3-manifolds into Euclidean 3-space). For
such surfaces total curvature consists of a sum of two-dimensional and one-dimensional integrals, which have various lower
bounds. Large sets of LT-surfaces which realize equality are then constructed. As an application, the orthogonal projection
of an immersed tight hypersurface in Euclidean 4-space is shown to have LT-tight critical image, and several related inequalities
are given.
Mathematics Subject Classifications (2000): 57N65, 14P99, 53C21, 53B25, 53B20. 相似文献
3.
In this article, we begin a systematic study of conformal properties of codimension-1 foliations. We first define and study
local conformal invariants. A case of particular interest is that of harmonic foliations of the plane. Then we study existence
of totally umbilical and “Dupin” foliations on compact 3-manifolds of constant curvature.
相似文献
4.
We show that in each dimension n = 4k, k≥ 2, there exist infinite sequences of closed simply connected Riemannian n-manifolds with nonnegative sectional curvature and mutually distinct oriented cobordism type.
W. Tuschmann’s research was supported in part by a DFG Heisenberg Fellowship. 相似文献
5.
N. Blažić M. Prvanović 《Abhandlungen aus dem Mathematischen Seminar der Universit?t Hamburg》2001,71(1):35-47
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. 相似文献
6.
Bao Fu Wang 《数学学报(英文版)》2009,25(8):1353-1362
Given a bounded convex domain Ω with C∞ boundary and a function ψ∈C∞(δΩ), Li-Simon-Chen can construct an Euclidean complete and W-complete convex hypersurface M with constant affine Gauss-Kronecker curvature, and they guess the M is also affine complete. In this paper, we give a confirmation answer. 相似文献
7.
8.
Ronaldo F. de Lima 《Annals of Global Analysis and Geometry》2001,20(4):325-343
Maximum principles at infinity generalize Hopf's maximum principle for hypersurfaces with constant mean curvature in R
n
. We establish such a maximum principle for parabolic surfaces in R3 with nonzero constant mean curvature and bounded Gaussian curvature. 相似文献
9.
DaGang Yang 《Inventiones Mathematicae》2000,142(2):435-450
An Einstein metric with positive scalar curvature on a 4-manifold is said to be normalized if Ric=1. A basic problem in Riemannian geometry is to classify Einstein 4-manifolds with positive sectional curvature in the category
of either topology, diffeomorphism, or isometry. It is shown in this paper that if the sectional curvature K of a normalized Einstein 4-manifold M satisfies the lower bound K≥ε0, ε0≡(-23)/120≈0.102843, or condition (b) of Theorem 1.1, then it is isometric to either S
4, RP
4 with constant sectional curvature K=1/3, or CP
2 with the normalized Fubini-Study metric. As a consequence, both the normalized moduli spaces of Einstein metrics which satisfy
either one of the above two conditions on S
4 and CP
2 contain only a single point. In particular, if M is a smooth 4-manifold which is homeomorphic to either S
4, RP
4, or CP
2 but not diffeomorphic to any of the three manifolds, then it can not support any normalized Einstein metric which satisfies
either one of the conditions.
Oblatum 4-II-1999 & 4-V-2000?Published online: 16 August 2000 相似文献
10.
Consider oriented surfaces immersed in
. Associated to them,
here are studied pairs of transversal foliations with
singularities, defined on the Elliptic region, where the
Gaussian curvature
, given
by the product of the principal curvatures
k
1,
k
2 is
positive. The leaves of the foliations are the
lines of harmonic mean
curvature, also called characteristic or
diagonal lines, along which
the normal curvature of the immersion is given by
, where
is the
arithmetic mean curvature. That is,
is the harmonic mean of the
principal curvatures k
1,
k
2 of
the immersion. The singularities of the foliations are the
umbilic points and
parabolic curves, where
k
1 =
k
2 and
, respectively.Here are determined the structurally stable patterns of
harmonic mean curvature lines
near the umbilic points, parabolic
curves and harmonic mean
curvature cycles, the periodic leaves of the
foliations. The genericity of these patterns is
established.This provides the three essential local ingredients to
establish sufficient conditions, likely to be also necessary,
for Harmonic Mean Curvature Structural
Stability of immersed surfaces. This study, outlined
towards the end of the paper, is a natural analog and complement
for that carried out previously by the authors for the
Arithmetic Mean Curvature and
the Asymptotic Structural
Stability of immersed surfaces, [13, 14, 17], and
also extended recently to the case of the
Geometric Mean Curvature
Configuration [15].The first author was partially supported by FUNAPE/UFG.
Both authors are fellows of CNPq.
This work was done under the project PRONEX/FINEP/MCT -
Conv. 76.97.1080.00 - Teoria Qualitativa das Equações Diferenciais
Ordinárias and CNPq - Grant 476886/2001-5. 相似文献
11.
We prove a strengthenedC
r
-closing lemma (r≥1) for wandering chain recurrent trajectories of flows without equilibrium states on the two-dimensional torus and for wandering
chain recurrent orbits of a diffeomorphism of the circle. The strengthenedC
r
-closing lemma (r≥1) is proved for a special class of infinitely smooth actions of the integer lattice ℤ
k
on the circle. The result is applied to foliations of codimension one with trivial holonomy group on the three-dimensional
torus.
Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 323–331, March, 1997.
Translated by S. K. Lando 相似文献
12.
We show that the complexity of a parabolic or conic spline approximating a sufficiently smooth curve with non-vanishing curvature
to within Hausdorff distance ɛ is c
1ɛ−1/4 + O(1), if the spline consists of parabolic arcs, and c
2ɛ−1/5 + O(1), if it is composed of general conic arcs of varying type. The constants c
1 and c
2 are expressed in the Euclidean and affine curvature of the curve. We also show that the Hausdorff distance between a curve
and an optimal conic arc tangent at its endpoints is increasing with its arc length, provided the affine curvature along the
arc is monotone. This property yields a simple bisection algorithm for the computation of an optimal parabolic or conic spline.
The research of SG and GV was partially supported by grant 6413 of the European Commission to the IST-2002 FET-Open project
Algorithms for Complex Shapes in the Sixth Framework Program. 相似文献
13.
We will simplify earlier proofs of Perelman’s collapsing theorem for 3-manifolds given by Shioya–Yamaguchi (J. Differ. Geom.
56:1–66, 2000; Math. Ann. 333: 131–155, 2005) and Morgan–Tian ( [math.DG], 2008). A version of Perelman’s collapsing theorem states: “Let
{M3i}\{M^{3}_{i}\}
be a sequence of compact Riemannian 3-manifolds with curvature bounded from below by (−1) and
$\mathrm{diam}(M^{3}_{i})\ge c_{0}>0$\mathrm{diam}(M^{3}_{i})\ge c_{0}>0
. Suppose that all unit metric balls in
M3iM^{3}_{i}
have very small volume, at most
v
i
→0 as
i→∞, and suppose that either
M3iM^{3}_{i}
is closed or has possibly convex incompressible toral boundary. Then
M3iM^{3}_{i}
must be a graph manifold for sufficiently large
i”. This result can be viewed as an extension of the implicit function theorem. Among other things, we apply Perelman’s critical
point theory (i.e., multiple conic singularity theory and his fibration theory) to Alexandrov spaces to construct the desired
local Seifert fibration structure on collapsed 3-manifolds. 相似文献
14.
Miguel Ortega Juan de Dios Pérez Young Jin Suh 《Czechoslovak Mathematical Journal》2006,56(2):377-388
In this paper we classify real hypersurfaces with constant totally real bisectional curvature in a non flat complex space
form M
m
(c), c ≠ 0 as those which have constant holomorphic sectional curvature given in [6] and [13] or constant totally real sectional
curvature given in [11]. 相似文献
15.
A. V. Potepun 《Journal of Mathematical Sciences》2006,139(2):6457-6478
It is well known that one can integrate any compactly supported, continuous, differential n-form over n-dimensional C1-manifolds in ℝm (m ≥ n). For n = 1, the integral may be defined over any locally rectifiable curve. Another generalization is the theory
of currents (linear functionals on the space of compactly supported C∞-differential forms). The theme of the article is integration of measurable differential n-forms over n-dimensional n C0-manifolds in ℝm with locally-finite n-dimensional variations (a generalization of locally rectifiable curves to dimension n > 1). The main
result states that any such manifold generates an n-dimensional current in ℝm (i.e., any compactly supported C∞ n-form can be integrated over a manifold with the properties mentioned above). Bibliography: 8 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 168–206. 相似文献
16.
We address the question: how large is the family of complete metricswith nonnegative sectional curvature on S 2 × R3? We classify theconnection metrics, and give several examples of nonconnection metrics.We provide evidence that the family is small by proving some rigidityresults for metrics more general than connection metrics.
相似文献17.
With regards to certain Riemannian foliations we consider Kasparov pairings of leafwise and transverse Dirac operators. Relative to a pairing with a transversal class we commence by establishing an index formula for foliations with leaves of nonpositive sectional curvature. The underlying ideas are then developed in a more general setting leading to pairings of images under the Baum-Connes map in geometricK-theory with transversal classes. Several ideas implicit in the work of Connes and Hilsum-Skandalis are formulated in the context of Riemannian foliations. From these we establish the notion of a dual pairing inK-homology and a theorem of the Grothendieck-Riemann-Roch type.R. G. D. was supported by The National Science Foundation under Grant No. DMS-9304283.J. F. G. and F. W. K. were supported in part by The National Science Foundation under Grant No. DMS-9208182.F. W. K. was also supported in part by an Arnold O. Beckman Research Award from the Research Board of the University of Illinois. 相似文献
18.
Maria Falcitelli 《Mediterranean Journal of Mathematics》2010,7(1):19-36
An algebraic characterization of generalized Sasakian-space-forms is stated. Then, one studies the almost contact metric manifolds
which are locally conformal to C
6-manifolds, simply called l.c. C
6-manifolds. In dimension 2n + 1 ≥ 5, any of these manifolds turns out to be locally conformal cosymplectic or globally conformal to a Sasakian manifold.
Curvature properties of l.c. C
6-manifolds are obtained, with particular attention to the k-nullity condition. This allows one to state a local classification theorem, in dimension 2n + 1 ≥ 5, under the hypothesis of constant sectional curvature. Moreover, one proves that an l.c. C
6–manifold is a generalized Sasakian-space-form if and only if it satisfies the k-nullity condition and has pointwise constant j{\varphi}-sectional curvature. Finally, local classification theorems for the generalized Sasakian-space-forms in the considered class
are obtained. 相似文献
19.
A Sasakian structure
=(\xi,\eta,\Phi,g) on a manifold Mis called positiveif its basic first Chern class c1(
) can be represented by a positive (1,1)-form with respect to its transverse holomorphic CR-structure. We prove a theorem that says that every positive Sasakian structure can be deformed to a Sasakian structure whose metric has positive Ricci curvature. This provides us with a new technique for proving the existence of positive Ricci curvature metrics on certain odd dimensional manifolds. As an example we give a completely independent proof of a result of Sha and Yang that for every nonnegative integer kthe 5-manifolds k#(S
2×S
3) admits metrics of positive Ricci curvature. 相似文献
20.
Jiguang Bao 《中国科学A辑(英文版)》1998,41(10):1047-1050
Under the mild conditions, it is proved that the convex surface is global C1.1, with the given Gaussian curvature 0≤K ∈ C
0
∞
and the given boundary curve. Examples are given to show that the regularity is optimal.
Project supported by the Doctoral Funds of China and the National Natural Science Foundation of China (Grant No. 19771009). 相似文献