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1.
   Abstract. The Upper Bound Conjecture is verified for a class of odd-dimensional simplicial complexes that in particular includes all Eulerian simplicial complexes with isolated singularities. The proof relies on a new invariant of simplicial complexes—a short simplicial h -vector.  相似文献   

2.
Abstract. The Upper Bound Conjecture is verified for a class of odd-dimensional simplicial complexes that in particular includes all Eulerian simplicial complexes with isolated singularities. The proof relies on a new invariant of simplicial complexes—a short simplicial h -vector.  相似文献   

3.
The angle defect, which is the standard way to measure the curvatures at the vertices of polyhedral surfaces, goes back at least as far as Descartes. Although the angle defect has been widely studied, there does not appear to be in the literature an axiomatic characterization of the angle defect. In this paper a characterization of the angle defect for simplicial surfaces is given, and it is shown that variants of the same characterization work for two known approaches to generalizing the angle defect to arbitrary 2-dimensional simplicial complexes. Simultaneously, a characterization of the Euler characteristic for 2-dimensional simplicial complexes is given in terms of being geometrically locally determined.  相似文献   

4.
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):ik-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   

5.
For a finitely triangulated closed surface M 2, let αx be the sum of angles at a vertex x. By the well-known combinatorial version of the 2- dimensional Gauss-Bonnet Theorem, it holds Σx(2π - αx) = 2πχ(M 2), where χ denotes the Euler characteristic of M 2, αx denotes the sum of angles at the vertex x, and the sum is over all vertices of the triangulation. We give here an elementary proof of a straightforward higher-dimensional generalization to Euclidean simplicial complexes K without assuming any combinatorial manifold condition. First, we recall some facts on simplicial complexes, the Euler characteristics and its local version at a vertex. Then we define δ(τ) as the normed dihedral angle defect around a simplex τ. Our main result is Στ (-1)dim(τ)δ(τ) = χ(K), where the sum is over all simplices τ of the triangulation. Then we give a definition of curvature κ(x) at a vertex and we prove the vertex-version Σ xK0 κ(x) = χ(K) of this result. It also possible to prove Morse-type inequalities. Moreover, we can apply this result to combinatorial (n + 1)-manifolds W with boundary B, where we prove that the difference of Euler characteristics is given by the sum of curvatures over the interior of W plus a contribution from the normal curvature along the boundary B:
$$\chi \left( W \right) - \frac{1}{2}\chi \left( B \right) = \sum {_{\tau \in W - B}} {\left( { - 1} \right)^{\dim \left( \tau \right)}} + \sum {_{\tau \in B}} {\left( { - 1} \right)^{\dim \left( \tau \right)}}\rho \left( \tau \right)$$
.
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6.
Let G be a powerful finite p-group. In this note, we give a short elementary proof of the following facts for all i ≥ 0: (i) exp Ωi(G) ≤ p i for odd p, and expΩi(G) ≤ 2 i+1 for p = 2; (ii) the index |G: G p i| coincides with the number of elements of G of order at most p i. Supported by the Spanish Ministry of Science and Education, grant MTM2004-04665, partly with FEDER funds, and by the University of the Basque Country, grant UPV05/99.  相似文献   

7.
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   

8.
We study a class of nonlinear evolutionary equations generated by an elliptic pseudo-differential operator, and with nonlinearity of the form G(u x ) where cη2 ≤ G(η) ≤ Cη2 for large |η|. For the evolution in spaces of periodic functions with zero mean we demonstrate existence of a universal absorbing set and compact attractor. Furthermore, we show that the attractor is of a finite Hausdorf dimension. The dissipation mechanism for the class of equations studied in the paper is akin to the nonlinear saturation in the Kuramoto-Sivashinsky equation. A similar generalization of the Kuramoto-Sivashinsky equation was studied by Nicolaenko et al. under the assumption of a purely quadratic nonlinearity and reflection invariance of both: the equation and solutions.   相似文献   

9.
We considered in Example 3.1 of the paper [1] an S-structure on R2n+s . We concluded that when s > 1 this manifold cannot be of constant φ-sectional curvature. Unfortunately this result is wrong. In fact, essentially due to a sign mistake in defining the φ-structure and a consequent transposition of the elements of the φ-basis (3.2), some of the Christoffel’s symbols were incorrect. In the present rectification, using a more slendler tecnique, we prove that our manifold is of constant φ-sectional curvature −3s and then it is η-Einstein.  相似文献   

10.
LetK=K 1,...,Kn be a family ofn convex sets inR d . For 0≦i<n denote byf i the number of subfamilies ofK of sizei+1 with non-empty intersection. The vectorf(K) is called thef-vectors ofK. In 1973 Eckhoff proposed a characterization of the set off-vectors of finite families of convex sets inR d by a system of inequalities. Here we prove the necessity of Eckhoff's inequalities. The proof uses exterior algebra techniques. We introduce a notion of generalized homology groups for simplicial complexes. These groups play a crucial role in the proof, and may be of some independent interest.  相似文献   

11.
It is proved that if a compact manifold admits a smooth action by a compact, connected, non-abelian Lie group, then it admits a metric of positive scalar curvature. This result is used to prove that if ∑ n is an exoticn-sphere which does not bound a spin manifold, then the only possible compact connected transformation groups of ∑ n are tori of dimension ≤[(n+1)/2]. Research partially supported by the Sloan Foundation and NSF Grant GP-34785X.  相似文献   

12.
Let X be a simplicial complex with ground set V. Define its Alexander dual as the simplicial complex X *={σVVσ X}. The combinatorial Alexander duality states that the ith reduced homology group of X is isomorphic to the (|V|−i−3)th reduced cohomology group of X * (over a given commutative ring R). We give a self-contained proof from first principles accessible to a nonexpert.  相似文献   

13.
Using elementary comparison geometry, we prove: Let (M, g) be a simply-connected complete Riemannian manifold of dimension ≥ 3. Suppose that the sectional curvature K satisfies −1 − s(r) ≤ K ≤ −1, where r denotes distance to a fixed point in M. If lim r → ∞ e2r s(r) = 0, then (M, g) has to be isometric to ℍ n . The same proof also yields that if K satisfies −s(r) ≤ K ≤ 0 where lim r → ∞ r 2 s(r) = 0, then (M, g) is isometric to ℝ n , a result due to Greene and Wu. Our second result is a local one: Let (M, g) be any Riemannian manifold. For a ∈ ℝ, if Ka on a geodesic ball B p (R) in M and K = a on ∂B p (R), then K = a on B p (R).  相似文献   

14.
In a 1967 paper, Banchoff described a theory of critical points and curvature for polyhedra embedded in Euclidean space. For each convex cell complex K in , and for each linear map satisfying a simple generality criterion, he defined an index for each vertex of K with respect to the map h, and showed that these indices satisfy two properties: (1) for each map h, the sum of the indices at all the vertices of K equalsχK and (2) for each vertex of K, the integral of the indices of the vertex with respect to all such linear maps equals the standard polyhedral notion of curvature of K at the vertex. In a previous paper, the author defined a different approach to curvature for arbitrary simplicial complexes, based upon a more direct generalization of the angle defect. In the present paper we present an analog of Banchoff ’s theory that works with our generalized angle defect.  相似文献   

15.
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 (kμ)-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   

16.
We show that if a closed manifold M admits an ℱ-structure (not necessarily polarized, possibly of rank zero) then its minimal entropy vanishes. In particular, this is the case if M admits a non-trivial S 1-action. As a corollary we obtain that the simplicial volume of a manifold admitting an ℱ-structure is zero.?We also show that if M admits an ℱ-structure then it collapses with curvature bounded from below. This in turn implies that M collapses with bounded scalar curvature or, equivalently, its Yamabe invariant is non-negative.?We show that ℱ-structures of rank zero appear rather frequently: every compact complex elliptic surface admits one as well as any simply connected closed 5-manifold.?We use these results to study the minimal entropy problem. We show the following two theorems: suppose that M is a closed manifold obtained by taking connected sums of copies of S 4, ℂP 2, 2,S 2×S 2and the K3 surface. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S 4,ℂP 2,S 2×S 2,ℂP 2#  2 or ℂP 2# ℂP 2. Finally, suppose that M is a closed simply connected 5-manifold. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S 5,S 3×S 2, then on trivial S 3-bundle over S 2 or the Wu-manifold SU(3)/SO(3). Oblatum 13-III-2002 & 12-VIII-2002?Published online: 8 November 2002 G.P. Paternain was partially supported by CIMAT, Guanajuato, México.?J. Petean is supported by grant 37558-E of CONACYT.  相似文献   

17.
In this paper, we consider the symmetric part of the so-called ith right shift operator. We determine its eigenvalues as also the associated eigenvectors in a complete and closed form. The proposed proof is elementary, using only basical skills such as Trigonometry, Arithmetic and Linear algebra. The first section is devoted to the introduction of the tackled problem. Second and third parts contain almost all the “technical” stuff of the proof. Afterwards, we continue with the end of the proof, provide a graphical illustration of the results, as well as an application on the polyhedral “sandwiching” of a special compact of arising in Signal theory.  相似文献   

18.
In this paper we show that there exists a unique local smooth solution for the Cauchy problem of the inhomogeneous Schr?dinger flow for maps from a compact Riemannian manifold M with dim(M) ≤ 3 into a compact K?hler manifold (N, J) with nonpositive Riemannian sectional curvature Received November 1, 1999, Revised January 14, 2000, Accepted March 29, 2000  相似文献   

19.
We prove that mirror nonsingular configurations of m points and n lines in ℝP 3 exist only for m≤3, n≡0 or 1 (mod 4) and for m=0 or 1 (mod 4), n≡0 (mod 2). In addition, we give an elementary proof of V. M. Kharlamov’s well-known result saying that if a nonsingular surface of degree four in ℝP 3 is noncontractible and has M≥5 components, then it is nonmirror. For the cases M=5, 6, 7 and 8, Kharlamov suggested an elementary proof using an analogy between such surfaces and configurations of M−1 points and a line. Our proof covers the remaining cases M=9, 10. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 299–308. Translated by N. Yu. Netsvetaev.  相似文献   

20.
Summary We consider a genaralization of contact metric manifolds given by assignment of 1-formsη1, . . . ,ηsand a compatible metric gon a manifold. With some integrability conditions they are called almost<span style='font-size:10.0pt;font-family:"Monotype Corsiva"; mso-bidi-font-family:"Monotype Corsiva"'>S-manifolds. We give a sufficient condition regarding the curvature of an almost<span style='font-size:10.0pt;font-family:"Monotype Corsiva";mso-bidi-font-family: "Monotype Corsiva"'>S-manifold to be locally isometric to a product of a Euclidean space and a sphere.  相似文献   

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