共查询到20条相似文献,搜索用时 31 毫秒
1.
Larry Guth 《Geometriae Dedicata》2006,123(1):113-129
These notes cover some of the main results of Gromov’s paper Filling Riemannian manifolds. The goal of these notes is to make
the results and proofs accessible to more people. The main result is that if (M,g) is a Riemannian manifold of dimension n, then there is a non-contractible curve in (M,g) of length at most C
n
Vol(M,g)1/n
. 相似文献
2.
B. Ammann 《Journal of Functional Analysis》2006,235(2):377-412
Let (M,g) be a compact Riemannian manifold of dimension n?3. We define the second Yamabe invariant as the infimum of the second eigenvalue of the Yamabe operator over the metrics conformal to g and of volume 1. We study when it is attained. As an application, we find nodal solutions of the Yamabe equation. 相似文献
3.
S��rgio de Moura Almaraz 《Calculus of Variations and Partial Differential Equations》2011,41(3-4):341-386
Let (M n , g) be a compact Riemannian manifold with boundary ?M. This paper is concerned with the set of scalar-flat metrics which are in the conformal class of g and have ?M as a constant mean curvature hypersurface. We prove that this set is compact for dimensions n ?? 7 under the generic condition that the trace-free 2nd fundamental form of ?M is nonzero everywhere. 相似文献
4.
5.
Shouwen Fang 《Archiv der Mathematik》2013,100(2):179-189
In the paper we consider a closed Riemannian manifold M with a time-dependent Riemannian metric g ij (t) evolving by ? t g ij = ?2S ij , where S ij is a symmetric two-tensor on (M,g(t)). We prove some differential Harnack inequalities for positive solutions of heat equations with potentials on (M,g(t)). Some applications of these inequalities will be obtained. 相似文献
6.
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
7.
R. Albuquerque 《Journal of Geometry》2014,105(2):327-342
We prove a Theorem on homotheties between two given tangent sphere bundles S r M of a Riemannian manifold M, g of \({{\rm dim}\geq3}\) , assuming different variable radius functions r and weighted Sasaki metrics induced by the conformal class of g. New examples are shown of manifolds with constant positive or with constant negative scalar curvature which are not Einstein. Recalling results on the associated almost complex structure I G and symplectic structure \({\omega^G}\) on the manifold TM, generalizing the well-known structure of Sasaki by admitting weights and connections with torsion, we compute the Chern and the Stiefel–Whitney characteristic classes of the manifolds TM and S r M. 相似文献
8.
Misha Gromov 《Central European Journal of Mathematics》2014,12(8):1109-1156
Groping our way toward a theory of singular spaces with positive scalar curvatures we look at the Dirac operator and a generalized Plateau problem in Riemannian manifolds with corners. Using these, we prove that the set of C 2-smooth Riemannian metrics g on a smooth manifold X, such that scal g (x) ≥ κ(x), is closed under C 0-limits of Riemannian metrics for all continuous functions κ on X. Apart from that our progress is limited but we formulate many conjectures. All along, we emphasize geometry, rather than topology of manifolds with their scalar curvatures bounded from below. 相似文献
9.
Asma Hassannezhad 《Journal of Functional Analysis》2011,261(12):3419-3436
In this paper, we find upper bounds for the eigenvalues of the Laplacian in the conformal class of a compact Riemannian manifold (M,g). These upper bounds depend only on the dimension and a conformal invariant that we call “min-conformal volume”. Asymptotically, these bounds are consistent with the Weyl law and improve previous results by Korevaar and Yang and Yau. The proof relies on the construction of a suitable family of disjoint domains providing supports for a family of test functions. This method is interesting for itself and powerful. As a further application of the method we obtain an upper bound for the eigenvalues of the Steklov problem in a domain with C1 boundary in a complete Riemannian manifold in terms of the isoperimetric ratio of the domain and the conformal invariant that we introduce. 相似文献
10.
R.Z. Domiaty 《Topology and its Applications》1985,20(1):39-46
Suppose that (M, g) and (M′, g′) are Lorentz manifolds, and that f: M → M′ is a bijection, such that f and f-1 preserve spacelike paths (f: M → M′ has this property, if for any spacelike path γ: J → M in (M ,g), the composition fγ: J → M′ is a spacelike path in (M′, g′)). Then f is a (manifold-) homeomorphism.This statement is the ‘spacelike’ version of an analogous ‘timelike’ theorem (Hawking, King and McCarthy [6] and Göbel [2] for strongly causal, and Malament [10] for general Lorentz manifolds).With this result it is possible to prove a conjecture of Göbel [3] which states that every bijection between time-orientable n-dimensional (n ? 3) Lorentz manifolds which preserves spacelike paths is a conformal C∞-diffeomorphism. 相似文献
11.
Ezequiel R. Barbosa Marcos Montenegro 《Journal of Mathematical Analysis and Applications》2008,344(2):699-702
Let (M,g) be a 4-dimensional compact Riemannian manifold and let a,f be positive smooth functions on M. In this note, we prove that the problem Δgu+a(x)u=f(x)u3 always admits a positive solution, up to a conformal deformation of g. This leads to a geometric obstruction result for the prescribed scalar curvature problem. 相似文献
12.
Fabio Podestá 《Results in Mathematics》1995,27(1-2):105-112
The aim of this paper is to generalize the construction of an Ambrose-Singer connection for Riemannian homogeneous manifolds to the case of cohomogeneity one Riemannian manifolds. Necessary and sufficient conditions are given on a Riemannian manifold (M,g) in order that there exists a Lie group of isometries acting on M with principal orbits of codimension one. 相似文献
13.
Ryszard Deszcz Ma?gorzata G?ogowska Miroslava Petrovi?-Torga?ev Leopold Verstraelen 《Results in Mathematics》2011,59(3-4):401-413
Chen ideal submanifolds M n in Euclidean ambient spaces E n+m (of arbitrary dimensions n ?? 2 and codimensions m??? 1) at each of their points do realise an optimal equality between their squared mean curvature, which is their main extrinsic scalar valued curvature invariant, and their ???C(= ??(2)?C) curvature of Chen, which is one of their main intrinsic scalar valued curvature invariants. From a geometric point of view, the pseudo-symmetric Riemannian manifolds can be seen as the most natural symmetric spaces after the real space forms, i.e. the spaces of constant Riemannian sectional curvature. From an algebraic point of view, the Roter manifolds can be seen as the Riemannian manifolds whose Riemann?CChristoffel curvature tensor R has the most simple expression after the real space forms, the latter ones being characterisable as the Riemannian spaces (M n , g) for which the (0, 4) tensor R is proportional to the Nomizu?CKulkarni square of their (0, 2) metric tensor g. In the present article, for the class of the Chen ideal submanifolds M n of Euclidean spaces E n+m , we study the relationship between these geometric and algebraic generalisations of the real space forms. 相似文献
14.
Let (M,g) be a complete simply-connected Riemannian manifold with nonpositive curvature, k its scalar curvature, and K a smooth function on M. We obtain a nonexistence result of complete metrics on M conformal to g and with K as their scalar curvature. 相似文献
15.
Claudio Buzzanca 《Rendiconti del Circolo Matematico di Palermo》1986,35(2):244-260
Characterisation of 2-dimensional Riemannian manifolds (M, g) (in particular, of surfaces with constant gaussian curvatureK=1/c 2, o,?1/c 2, respectively) whose tangent circle bundle (T cM, gs) (g s=Sasaki metric) admit an «almost-regular» vector field belonging to an eigenspace of the Ricci operator. 相似文献
16.
A Riemannian g.o. manifold is a homogeneous Riemannian manifold (M,g) on which every geodesic is an orbit of a one-parameter group of isometries. It is known that every simply connected Riemannian g.o. manifold of dimension ?5 is naturally reductive. In dimension 6 there are simply connected Riemannian g.o. manifolds which are in no way naturally reductive, and their full classification is known (including compact examples). In dimension 7, just one new example has been known up to now (namely, a Riemannian nilmanifold constructed by C. Gordon). In the present paper we describe compact irreducible 7-dimensional Riemannian g.o. manifolds (together with their “noncompact duals”) which are in no way naturally reductive. 相似文献
17.
Let M be a complete Riemannian manifold possibly with a boundary?M.For any C~1-vector field Z,by using gradient/functional inequalities of the(reflecting)diffusion process generated by L:=?+Z,pointwise characterizations are presented for the Bakry-Emery curvature of L and the second fundamental form of?M if it exists.These characterizations extend and strengthen the recent results derived by Naber for the uniform norm‖RicZ‖∞on manifolds without boundaries.A key point of the present study is to apply the asymptotic formulas for these two tensors found by the first author,such that the proofs are significantly simplified. 相似文献
18.
José F. Escobar 《Journal of Functional Analysis》2003,202(2):424-442
Let (Mn,g) be a compact manifold with boundary with n?2. In this paper we discuss uniqueness and non-uniqueness of metrics in the conformal class of g having the same scalar curvature and the mean curvature of the boundary of M. 相似文献
19.
Abdlkadir ZDEGER 《数学学报(英文版)》2013,29(2):373-382
It is well known that the Einstein tensor G for a Riemannian manifold defined by G βα = R βα 1/2 Rδβα , R βα = g βγ R γα where R γα and R are respectively the Ricci tensor and the scalar curvature of the manifold, plays an important part in Einstein's theory of gravitation as well as in proving some theorems in Riemannian geometry. In this work, we first obtain the generalized Einstein tensor for a Weyl manifold. Then, after studying some properties of generalized Einstein tensor, we prove that the conformal invariance of the generalized Einstein tensor implies the conformal invariance of the curvature tensor of the Weyl manifold and conversely. Moreover, we show that such Weyl manifolds admit a one-parameter family of hypersurfaces the orthogonal trajectories of which are geodesics. Finally, a necessary and sufficient condition in order that the generalized circles of a Weyl manifold be preserved by a conformal mapping is stated in terms of generalized Einstein tensors at corresponding points. 相似文献
20.
Nicolas Saintier 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(2):689-703
Let (M,g) be a smooth compact Riemannian manifold. We first give the value of the best first constant for the critical embedding H2(M)?L2?(M) for second-order Sobolev spaces of functions invariant by some subgroup of the isometry group of (M,g). We also prove that we can take ?=0 in the corresponding inequality under some geometric assumptions. As an application we give a sufficient condition for the existence of a smooth positive symmetric solution to a critical equation with a symmetric Paneitz-Branson-type operator. A sufficient condition for the existence of a nodal solution to such an equation is also derived. We eventually prove a multiplicity result for such an equation. 相似文献