共查询到20条相似文献,搜索用时 15 毫秒
1.
Brian Clarke 《Calculus of Variations and Partial Differential Equations》2010,39(3-4):533-545
We prove that the L 2 Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold induces a metric space structure. As the L 2 metric is a weak Riemannian metric, this fact does not follow from general results. In addition, we prove several results on the exponential mapping and distance function of a weak Riemannian metric on a Hilbert/Fréchet manifold. The statements are analogous to, but weaker than, what is known in the case of a Riemannian metric on a finite-dimensional manifold or a strong Riemannian metric on a Hilbert manifold. 相似文献
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Let M~n(n ≥ 4) be an oriented closed submanifold with parallel mean curvature in an(n + p)-dimensional locally symmetric Riemannian manifold N~(n+p). We prove that if the sectional curvature of N is positively pinched in [δ, 1], and the Ricci curvature of M satisfies a pinching condition, then M is either a totally umbilical submanifold, or δ = 1, and N is of constant curvature. This result generalizes the geometric rigidity theorem due to Xu and Gu[15]. 相似文献
5.
Xiang Dong Li 《Probability Theory and Related Fields》2003,125(1):96-134
We introduce Sobolev spaces and capacities on the path space P m 0 (M) over a compact Riemannian manifold M. We prove the smoothness of the Itô map and the stochastic anti-development map in the sense of stochastic calculus of variation. We establish a Sobolev norm comparison theorem and a capacity comparison theorem between the Wiener space and the path space P m 0 (M). Moreover, we prove the tightness of (r, p)-capacities on P m 0 (M), \(\), which generalises a result due to Airault-Malliavin and Sugita on the Wiener space. Finally, we extend our results to the fractional Hölder continuous path space \(\). 相似文献
6.
Regina Rotman 《Topology》2007,46(4):343-356
In this paper we will estimate the smallest length of a minimal geodesic net on an arbitrary closed Riemannian manifold Mn in terms of the diameter of this manifold and its dimension. Minimal geodesic nets are critical points of the length functional on the space of immersed graphs into a Riemannian manifold. We prove that there exists a minimal geodesic net that consists of m geodesics connecting two points p,q∈Mn of total length ≤md, where m∈{2,…,(n+1)} and d is the diameter of Mn. We also show that there exists a minimal geodesic net with at most n+1 vertices and geodesic segments of total length .These results significantly improve one of the results of [A. Nabutovsky, R. Rotman, The minimal length of a closed geodesic net on a Riemannian manifold with a nontrivial second homology group, Geom. Dedicata 113 (2005) 234-254] as well as most of the results of [A. Nabutovsky, R. Rotman, Volume, diameter and the minimal mass of a stationary 1-cycle, Geom. Funct. Anal. 14 (4) (2004) 748-790]. 相似文献
7.
Gerd Kainz 《Monatshefte für Mathematik》1984,98(3):211-217
E. Binz [1] considered two canonical Riemannian metrics on the space of embeddings of a closed (n–1) dimensional manifold into
n
, and computed the geodesic sprays. Here we consider the space of immersions Imm (M, N) whereM is without boundary, and we compute the covariant derivative (in the form of its connector) and the Riemannian curvature of one of these metrics, the non trivial one. The setting is close to that used byP. Michor [2], and we refer the reader to this paper for notation. 相似文献
8.
Corneliu Ursescu 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(10):3295-3302
This paper concerns a near subtraction result for multifunctions with locally closed graph in metric spaces. 相似文献
9.
Jürgen Eichhorn 《Mathematische Nachrichten》2003,253(1):8-34
We introduce uniform structures of proper metric spaces and open Riemannian manifolds, characterize their (arc) components, present new invariants like e.g. Lipschitz and Gromov–Hausdorff cohomology, specialize to uniform triangulations of manifolds and prove that the presence of a spectral gap above zero is a bounded homotopy invariant. 相似文献
10.
S. Alhemedan 《Journal of Mathematical Analysis and Applications》2010,370(2):364-372
Polyharmonic functions are considered on open sets in a Riemannian manifold R and their potential-theoretic properties are studied using the notion of complete m-potentials. Also one obtains here some characterizations of domains in R on which such complete m-potentials exist. 相似文献
11.
A. Caminha 《Bulletin of the Brazilian Mathematical Society》2011,42(2):277-300
In this paper we examine different aspects of the geometry of closed conformal vector fields on Riemannian manifolds. We begin
by getting obstructions to the existence of closed conformal and nonparallel vector fields on complete manifolds with nonpositive
Ricci curvature, thus generalizing a theorem of T.K. Pan. Then we explain why it is so difficult to find examples, other than
trivial ones, of spaces having at least two closed, conformal and homothetic vector fields. We then focus on isometric immersions,
firstly generalizing a theorem of J. Simons on cones with parallel mean curvature to spaces furnished with a closed, Ricci
null conformal vector field; then we prove general Bernstein-type theorems for certain complete, not necessarily cmc, hypersurfaces
of Riemannian manifolds furnished with closed conformal vector fields. In particular, we obtain a generalization of theorems
J. Jellett and A. Barros and P. Sousa for complete cmc radial graphs over finitely punctured geodesic spheres of Riemannian
space forms. 相似文献
12.
Jing-xiao ZHANG 《应用数学学报(英文版)》2013,29(3):499-508
The flow on the Wiener space associated to a tangent process constructed by Cipriano and Cruzeiro, as well as by Gong and Zhang does not allow to recover Driver’s Cameron-Martin theorem on Riemannian path space. The purpose of this work is to refine the method of the modified Picard iteration used in the previous work by Gong and Zhang and to try to recapture and extend the result of Driver. In this paper, we establish the existence and uniqueness of a flow associated to a Cameron-Martin type vector field on the path space over a Riemannian manifold. 相似文献
13.
Jean-Pierre Ezin Marco Rigoli Isabel M. C. Salavessa 《Israel Journal of Mathematics》1990,71(2):193-209
We extend some rigidity results of Aleksandrov and Ros on compact hypersurfaces inR
n
to more general ambient spaces with the aid of the notion of almost conformal vector fields. These latter, at least locally,
always exist and allow us to find interesting integral formulas fitting our purposes. 相似文献
14.
Reiko Miyaoka 《Differential Geometry and its Applications》2013,31(1):130-139
We extend theorems of É. Cartan, Nomizu, Münzner, Q.M. Wang, and Ge–Tang on isoparametric functions to transnormal functions on a general Riemannian manifold. We show that if a complete Riemannian manifold M admits a transnormal function, then M is diffeomorphic to either a vector bundle over a submanifold, or a union of two disk bundles over two submanifolds. Moreover, a singular level set Q is austere and minimal, if exists, and generic level sets are tubes over Q. We give a criterion for a transnormal function to be an isoparametric function. 相似文献
15.
Curve shortening in a Riemannian manifold 总被引:1,自引:0,他引:1
In this paper, we study the curve shortening flow in a general Riemannian manifold. We have many results for the global behavior
of the flow. In particular, we show the following results: let M be a compact Riemannian manifold. (1) If the curve shortening flow exists for infinite time, and
, then for every n > 0,
. Furthermore, the limiting curve exists and is a closed geodesic in M. (2) In M × S
1, if γ0 is a ramp, then we have a global flow which converges to a closed geodesic in C
∞ norm. As an application, we prove the theorem of Lyusternik and Fet.
相似文献
16.
A. A. Murach 《Ukrainian Mathematical Journal》2007,59(6):874-893
We study linear elliptic pseudodifferential operators in the improved scale of functional Hilbert spaces on a smooth closed manifold. Elements of this scale are isotropic Hörmander-Volevich-Paneyakh spaces. We investigate the local smoothness of a solution of an elliptic equation in the improved scale. We also study elliptic pseudodifferential operators with parameter. 相似文献
17.
A note on closed images of locally compact metric spaces 总被引:1,自引:0,他引:1
Shou Lin 《Acta Mathematica Hungarica》2005,109(1-2):157-162
Summary A decomposition theorem about closed images of locally compact metric spaces is discussed. It is shown that a space is a closed
image of a locally compact metric space if and only if it is a regular Fréchet space with a point-countable k-network, and each of its closed first-countable subset is locally compact. 相似文献
18.
E. D. Rodionov 《Siberian Mathematical Journal》1991,32(3):455-459
Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 32, No. 3, pp. 126–131, May–June, 1991. 相似文献
19.
Shun Maeta 《Annals of Global Analysis and Geometry》2014,46(1):75-85
We consider biharmonic maps $\phi :(M,g)\rightarrow (N,h)$ from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. Assume that $ p $ satisfies $ 2\le p <\infty $ . If for such a $ p $ , $\int _M|\tau (\phi )|^{ p }\,\mathrm{d}v_g<\infty $ and $\int _M|\,\mathrm{d}\phi |^2\,\mathrm{d}v_g<\infty ,$ where $\tau (\phi )$ is the tension field of $\phi $ , then we show that $\phi $ is harmonic. For a biharmonic submanifold, we obtain that the above assumption $\int _M|\,\mathrm{d}\phi |^2\,\mathrm{d}v_g<\infty $ is not necessary. These results give affirmative partial answers to the global version of generalized Chen’s conjecture. 相似文献