共查询到20条相似文献,搜索用时 31 毫秒
1.
The Lichnerowicz conjecture asserted that every harmonic Riemannian manifold is locally isometric to a two-point homogeneous space. In 1992, E. Damek and F. Ricci produced a family of counter-examples to this conjecture, which arise as abelian extensions of two-step nilpotent groups of type-H. In this paper we consider a broader class of Riemannian manifolds: solvmanifolds of Iwasawa type with algebraic rank one and two-step nilradical. Our main result shows that the Damek–Ricci spaces are the only harmonic manifolds of this type. 相似文献
2.
Yau made the following conjecture: For a complete noncompact manifold with nonnegative Ricci curvature the space of harmonic
functions with polynomial growth of a fixed rate is finite dimensional. we extend the result on the Laplace operator to that
on the symmetric diffusion operator, and prove the space of L-harmonic functions with polynomial growth of a fixed rate is finite-dimensional, when m-dimensional Bakery-Emery Ricci curvature of the symmetric diffusion operator on the complete noncompact Riemannian manifold
is nonnegative. 相似文献
3.
We recall a curvature identity for 4-dimensional compact Riemannian manifolds as derived from the generalized Gauss–Bonnet formula. We extend this curvature identity to non-compact 4-dimensional Riemannian manifolds. We also give some applications of this curvature identity. 相似文献
4.
Sekigawa proved in 1977 that a 3-dimensional Riemannian manifold which is curvature homogeneous up to order 1 in the sense of I.M. Singer is always locally homogeneous. We deal here with the modification of the curvature homogeneity which is said to be “of type (1, 3)”. We give example of a 3-dimensional Riemannian manifold which is curvature homogeneous up to order 1 in the modified sense but still not locally homogeneous. 相似文献
5.
Ying Shen 《Proceedings of the American Mathematical Society》1997,125(3):901-905
In this paper, we borrowed some ideas from general relativity and find a Robinson-type identity for the overdetermined system of partial differential equations in the Fischer-Marsden conjecture. We proved that if there is a nontrivial solution for such an overdetermined system on a 3-dimensional, closed manifold with positive scalar curvature, then the manifold contains a totally geodesic 2-sphere.
6.
In this paper,we show that every harmonic map from a compact K?hler manifold with uniformly RC-positive curvature to a Riemannian manifold with non-positive complex sectional curvature is constant.In particular,there is no non-constant harmonic map from a compact Koahler manifold with positive holomorphic sectional curvature to a Riemannian manifold with non-positive complex sectional curvature. 相似文献
7.
Takeshi Isobe 《Journal of Geometric Analysis》1998,8(3):447-463
We study the regularity of harmonic maps from Riemannian manifold into a static Lorentzian manifold. We show that when the
domain manifold is two-dimensional, any weakly harmonic map is smooth. We also show that when dimension n of the domain manifold
is greater than two, there exists a weakly harmonic map for the Dirichlet problem which is smooth except for a closed set
whose (n − 2)-dimensional Hausdorff measure is zero. 相似文献
8.
Conformal harmonic maps from a 4-dimensional conformal manifold to a Riemannian manifold are maps satisfying a certain conformally invariant fourth order equation. We prove a general existence result for conformal harmonic maps, analogous to the Eells–Sampson theorem for harmonic maps. The proof uses a geometric flow and relies on results of Gursky–Viaclovsky and Lamm. 相似文献
9.
A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to be topologically formal. Nonetheless, we show that among the homogeneous Riemannian metrics of positive sectional curvature a geometrically formal metric is either symmetric, or a metric on a rational homology sphere. 相似文献
10.
We give an estimate of the smallest spectral value of the Laplace operator on a complete noncompact stable minimal hypersurface
M in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space,
we prove that if a complete minimal hypersurface M has sufficiently small total scalar curvature then M has only one end. We also obtain a vanishing theorem for L
2 harmonic 1-forms on minimal hypersurfaces in a Riemannian manifold with sectional curvature bounded below by a negative constant.
Moreover, we provide sufficient conditions for a minimal hypersurface in a Riemannian manifold with nonpositive sectional
curvature to be stable. 相似文献
11.
In this paper, we will introduce the notion of harmonic stability for complete minimal hypersurfaces in a complete Riemannian
manifold. The first result we prove, is that a complete harmonic stable minimal surface in a Riemannian manifold with non-negative
Ricci curvature is conformally equivalent to either a plane R
2 or a cylinder R × S
1, which generalizes a theorem due to Fischer-Colbrie and Schoen [12].
The second one is that an n ≥ 2-dimensional, complete harmonic stable minimal, hypersurface M in a complete Riemannian manifold with non-negative sectional curvature has only one end if M is non-parabolic. The third one, which we prove, is that there exist no non-trivial L
2-harmonic one forms on a complete harmonic stable minimal hypersurface in a complete Riemannian manifold with non-negative
sectional curvature. Since the harmonic stability is weaker than stability, we obtain a generalization of a theorem due to
Miyaoka [20] and Palmer [21].
Research partially Supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science
and Technology, Japan.
The author’s research was supported by grant Proj. No. KRF-2007-313-C00058 from Korea Research Foundation, Korea.
Authors’ addresses: Qing-Ming Cheng, Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga
840-8502, Japan; Young Jin Suh, Department of Mathematics, Kyungpook National University, Taegu 702-701, South Korea 相似文献
12.
Qilin Yang 《Differential Geometry and its Applications》2007,25(1):1-7
It is well known there is no non-constant harmonic map from a closed Riemannian manifold of positive Ricci curvature to a complete Riemannian manifold with non-positive sectional curvature. By reducing the assumption on the Ricci curvature to one on the scalar curvature, such vanishing theorem cannot hold in general. This raises the question: “What information can we obtain from the existence of non-constant harmonic map?” This paper gives answer to this problem; the results obtained are optimal. 相似文献
13.
Seungsu Hwang 《manuscripta mathematica》2000,103(2):135-142
It is well known that critical points of the total scalar curvature functional ? on the space of all smooth Riemannian structures
of volume 1 on a compact manifold M are exactly the Einstein metrics. When the domain of ? is restricted to the space of constant scalar curvature metrics, there
has been a conjecture that a critical point is also Einstein or isometric to a standard sphere. In this paper we prove that
n-dimensional critical points have vanishing n− 1 homology under a lower Ricci curvature bound for dimension less than 8.
Received: 12 July 1999 相似文献
14.
On an n-dimensional compact, orientable, connected Riemannian manifold, we consider the curvature operator acting on the space of covariant traceless symmetric 2-tensors. We prove that, if the curvature operator is negative, then the manifold admits no nonzero conformally Killing p-forms for p = 1, 2, …, n ? 1. On the other hand, we prove that the dimension of the vector space of conformally Killing p-forms on an n-dimensional compact simply-connected conformally flat Riemannian manifold (M,g) is not zero. 相似文献
15.
We prove global regularity for the solution to the Cauchy problem with regular data for an equivariant harmonic map from the 2 + 1-dimensional Minkowski space into a two-dimensional, rotationally symmetric, and geodesically convex Riemannian manifold. 相似文献
16.
Guo-liang Xu 《计算数学(英文版)》2003,(5)
In this paper, we provide simple and explicit formulas for computing Riemannian curvatures, mean curvature vectors, principal curvatures and principal directions for a 2-dimensional Riemannian manifold embedded in IRk with k > 3. 相似文献
17.
Guo-lian~Xu ChandrajitL.Bajaj 《计算数学(英文版)》2003,21(5):681-688
In this paper, we provide simple and explicit formulas for computing Riemannian curvatures, mean curvature vectors, principal curvatures and principal directions for a 2-dimensional Riemannian manifold embedded in IR^k with k ≥ 3. 相似文献
18.
《Indagationes Mathematicae》2019,30(4):542-552
In this paper, we study the impact of geodesic vector fields (vector fields whose trajectories are geodesics) on the geometry of a Riemannian manifold. Since, Killing vector fields of constant lengths on a Riemannian manifold are geodesic vector fields, leads to the question of finding sufficient conditions for a geodesic vector field to be Killing. In this paper, we show that a lower bound on the Ricci curvature of the Riemannian manifold in the direction of geodesic vector field gives a sufficient condition for the geodesic vector field to be Killing. Also, we use a geodesic vector field on a 3-dimensional complete simply connected Riemannian manifold to find sufficient conditions to be isometric to a 3-sphere. We find a characterization of an Einstein manifold using a Killing vector field. Finally, it has been observed that a major source of geodesic vector fields is provided by solutions of Eikonal equations on a Riemannian manifold and we obtain a characterization of the Euclidean space using an Eikonal equation. 相似文献
19.
The Liouville property of a complete Riemannian manifold M (i.e., the question whether there exist non-trivial bounded harmonic functions on M) attracted a lot of attention. For Cartan–Hadamard manifolds the role of lower curvature bounds is still an open problem.
We discuss examples of Cartan–Hadamard manifolds of unbounded curvature where the limiting angle of Brownian motion degenerates
to a single point on the sphere at infinity, but where nevertheless the space of bounded harmonic functions is as rich as
in the non-degenerate case. To see the full boundary the point at infinity has to be blown up in a non-trivial way. Such examples
indicate that the situation concerning the famous conjecture of Greene and Wu about existence of non-trivial bounded harmonic
functions on Cartan–Hadamard manifolds is much more complicated than one might have expected.
相似文献
20.
An old problem asks whether a Riemannian manifold can be isospectral to a Riemannian orbifold with nontrivial singular set. In this short note we show that under the assumption of Schanuel’s conjecture in transcendental number theory, this is impossible whenever the orbifold and manifold in question are length-commensurable compact locally symmetric spaces of nonpositive curvature associated to simple Lie groups. 相似文献