共查询到20条相似文献,搜索用时 31 毫秒
1.
本文研究和刻画了射影Ricci平坦的Kropina度量.利用Kropina度量的S-曲率和Ricci曲率的公式,得到了Kropina度量的射影Ricci曲率公式.在此基础上得到了Kropina度量是射影Ricci平坦度量的充分必要条件.进一步,作为自然的应用,本文研究和刻画了由一个黎曼度量和一个具有常数长度的Killing 1-形式定义的射影Ricci平坦的Kropina度量,也刻画了具有迷向S-曲率的射影Ricci平坦的Kropina度量.在这种情形下,Kropina度量是Ricci平坦度量. 相似文献
2.
Given a Riemannian metric on a compact smooth manifold, we consider its Schouten tensor, which is a tensor field of type (0, 2) arising in the remainder of the Weyl part in the standard decomposition of the curvature tensor of the metric. We study extremal properties of the Schouten functional, defined to be the scaling-invariant L 2-norm of the Schouten tensor. It is proved, for instance, that space form metrics are characterized as critical points of the Schouten functional among conformally flat metrics. 相似文献
3.
J.-M. Schlenker 《Geometric And Functional Analysis》2002,12(2):395-435
A classical theorem, mainly due to Aleksandrov [Al2] and Pogorelov [P], states that any Riemannian metric on S
2
with curvature K > —1 is induced on a unique convex surface in H
3
. A similar result holds with the induced metric replaced by the third fundamental form. We show that the same phenomenon
happens with yet another metric on immersed surfaces, which we call the horospherical metric.?This result extends in higher
dimensions, the metrics obtained are then conformally flat. One can also study equivariant immersions of surfaces or the metrics
obtained on the boundaries of hyperbolic 3-manifolds. Some statements which are difficult or only conjectured for the induced
metric or the third fundamental form become fairly easy when one considers the horospherical metric, which thus provides a
good boundary condition for the construction of hyperbolic metrics on a manifold with boundary.?The results concerning the
third fundamental form are obtained using a duality between H
3
and the de Sitter space . In the same way, the results concerning the horospherical metric are proved through a duality between H
n
and the space of its horospheres, which is naturally endowed with a fairly rich geometrical structure.
Submitted: March 2001, Revised: November 2001. 相似文献
4.
Benling Li 《Differential Geometry and its Applications》2013,31(6):718-724
In this paper, we study the locally dually flat Finsler metrics which arise from information geometry. An equivalent condition of locally dually flat Finsler metrics is given. We find a new method to construct locally dually flat Finsler metrics by using a projectively flat Finsler metric under the condition that the projective factor is also a Finsler metric. Finally, we find that many known Finsler metrics are locally dually flat Finsler metrics determined by some projectively flat Finsler metrics. 相似文献
5.
We study the mechanisms of the non properness of the action of
the group of diffeomorphisms on the space of Lorentzian metrics
of a compact manifold.
In particular, we prove that nonproperness entails the presence of
lightlike geodesic foliations of codimension 1.
On the 2-torus, we prove that a metric with constant curvature
along one of its lightlike foliation is actually flat. This
allows us to show that the restriction of the action to the set of
non-flat metrics is proper and that on the set of flat metrics of
volume 1 the action is ergodic.
Finally, we show that, contrarily to the Riemannian case, the
space of metrics without isometries is not always open. 相似文献
6.
We study the Lipschitz metric on a Teichmüller space (definedby Thurston) and compare it with the Teichmüller metric.We show that in the thin part of the Teichmüller spacethe Lipschitz metric is approximated up to a bounded additivedistortion by the sup-metric on a product of lower-dimensionalspaces (similar to the Teichmüller metric as shown by Minsky).In the thick part, we show that the two metrics are equal upto a bounded additive error. However, these metrics are notcomparable in general; we construct a sequence of pairs of pointsin the Teichmüller space, with distances that approachzero in the Lipschitz metric while they approach infinity inthe Teichmüller metric. 相似文献
7.
In this article, we investigate the balanced condition and the existence of an Engliš expansion for the Taub-NUT metrics on
\mathbbC2{\mathbb{C}^2} . Our first result shows that a Taub-NUT metric on
\mathbbC2{\mathbb{C}^2} is never balanced unless it is the flat metric. The second one shows that an Engliš expansion of the Rawnsley’s function
associated to a Taub-NUT metric always exists, while the coefficient a
3 of the expansion vanishes if and only if the Taub-NUT metric is indeed the flat one. 相似文献
8.
In this paper, we study the problem whether a Finsler metric of scalar flag curvature is locally projectively flat. We consider a special class of Finsler metrics — square metrics which are defined by a Riemannian metric and a 1-form on a manifold. We show that in dimension n ≥ 3, any square metric of scalar flag curvature is locally projectively flat. 相似文献
9.
In this paper, we consider a class of Finsler metrics which obtained by Kropina change of the class of generalized m-th root Finsler metrics. We classify projectively flat Finsler metrics in this class of metrics. Then under a condition, we show that every projectively flat Finsler metric in this class with constant flag curvature is locally Minkowskian. Finally, we find necessary and sufficient condition under which this class of metrics be locally dually flat. 相似文献
10.
Florent Balacheff 《Geometriae Dedicata》2006,121(1):61-71
We study the systolic area (defined as the ratio of the area over the square of the systole) of the 2-sphere endowed with
a smooth Riemannian metric as a function of this metric. This function, bounded from below by a positive constant over the
space of metrics, admits the standard metric g
0 as a critical point, although it does not achieve the conjectured global minimum: we show that for each tangent direction
to the space of metrics at g
0, there exists a variation by metrics corresponding to this direction along which the systolic area can only increase.
相似文献
11.
Inspired by the recent work [HHM03], we prove two stability results for compact Riemannian manifolds with nonzero parallel spinors. Our first result says that Ricci flat metrics which also admit nonzero parallel spinors are stable (in the direction of changes in conformal structures) as the critical points of the total scalar curvature functional. Our second result, which is a local version of the first one, shows that any metric of positive scalar curvature cannot lie too close to a metric with nonzero parallel spinor. We also prove a rigidity result for special holonomy metrics. In the case of SU(m) holonomy, the rigidity result implies that scalar flat deformations of Calabi-Yau metric must be Calabi-Yau. Finally we explore the connection with a positive mass theorem of [D03], which presents another approach to proving these stability and rigidity results. Dedicated to Jeff Cheeger for his sixtieth birthday 相似文献
12.
Olivier Biquard 《Inventiones Mathematicae》2002,148(3):545-607
A conformal metric on a 4-ball induces on the boundary 3-sphere a conformal metric and a trace-free second fundamental form.
Conversely, such a data on the 3-sphere is the boundary of a unique selfdual conformal metric, defined in a neighborhood of
the sphere. In this paper we characterize the conformal metrics and trace-free second fundamental forms on the 3-sphere (close
to the standard round metric) which are boundaries of selfdual conformal metrics on the whole 4-ball. When the data on the
boundary is reduced to a conformal metric (the trace-free part of the second fundamental form vanishes), one may hope to find
in the conformal class of the filling metric an Einstein metric, with a pole of order 2 on the boundary. We determine which
conformal metrics on the 3-sphere are boundaries of such selfdual Einstein metrics on the 4-ball. In particular, this implies
the Positive Frequency Conjecture of LeBrun. The proof uses twistor theory, which enables to translate the problem in terms
of complex analysis; this leads us to prove a criterion for certain integrable CR structures of signature (1,1) to be fillable
by a complex domain. Finally, we solve an analogous, higher dimensional problem: selfdual Einstein metrics are replaced by
quaternionic-K?hler metrics, and conformal structures on the boundary by quaternionic contact structures (previously introduced
by the author); in contrast with the 4-dimensional case, we prove that any small deformation of the standard quaternionic
contact structure on the (4m−1)-sphere is the boundary of a quaternionic-K?hler metric on the (4m)-ball.
Oblatum 29-XI-2000 & 7-XI-2001?Published online: 1 February 2002 相似文献
13.
Lixin Liu Athanase Papadopoulos Weixu Su Guillaume Théret 《Monatshefte für Mathematik》2010,31(1):295-311
We consider some metrics and weak metrics defined on the Teichmüller space of a surface of finite type with nonempty boundary,
that are defined using the hyperbolic length spectrum of simple closed curves and of properly embedded arcs, and we compare
these metrics and weak metrics with the Teichmüller metric. The comparison is on subsets of Teichmüller space which we call
“ε
0-relative e{\epsilon}-thick parts”, and whose definition depends on the choice of some positive constants ε
0 and e{\epsilon}. Meanwhile, we give a formula for the Teichmüller metric of a surface with boundary in terms of extremal lengths of families
of arcs. 相似文献
14.
D. B. McReynolds 《Geometriae Dedicata》2007,129(1):47-55
In this article we verify an orbifold version of a conjecture of Nimershiem from 1998. Namely, for every flat n-manifold M, we show that the set of similarity classes of flat metrics on M which occur as a cusp cross-section of a hyperbolic (n + 1)-orbifold is dense in the space of similarity classes of flat metrics on M. The set used for density is precisely the set of those classes which arise in arithmetic orbifolds.
相似文献
15.
For a given finite subset S of a compact Riemannian manifold (M,g) whose Schouten curvature tensor belongs to a given cone, we establish a necessary and sufficient condition for the existence and uniqueness of a conformal metric on M\S such that each point of S corresponds to an asymptotically flat end and that the Schouten tensor of the conformal metric belongs to the boundary of the given cone. As a by-product, we define a purely local notion of Ricci lower bounds for continuous metrics that are conformal to smooth metrics and prove a corresponding volume comparison theorem. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. 相似文献
16.
Using the heat kernel, we derive first a local Gauss–Bonnet–Chern theorem for manifolds with a non-product metric near the
boundary. Then we establish an anomaly formula for Ray–Singer metrics defined by a Hermitian metric on a flat vector bundle
over a Riemannian manifold with boundary, not assuming that the Hermitian metric on the flat vector bundle is flat nor that
the Riemannian metric has product structure near the boundary.
Received: January 2004; Revision: February 2005; Accepted: September 2005 相似文献
17.
Akbar Tayebi Mohammad Shahbazi Nia Esmaeil Peyghan 《Linear algebra and its applications》2012,437(2):675-683
In this paper, we characterize locally dually flat generalized m-th root Finsler metrics. Then we find a condition under which a generalized m-th root metric is projectively related to a m-th root metric. Finally, we prove that if a generalized m-th root metric is conformal to a m-th root metric, then both of them reduce to Riemannian metrics. 相似文献
18.
Our main goal in this work is to deal with results concern to the \(\sigma _2\)-curvature. First we find a symmetric 2-tensor canonically associated to the \(\sigma _2\)-curvature and we present an almost-Schur-type lemma. Using this tensor, we introduce the notion of \(\sigma _2\)-singular space and under a certain hypothesis we prove a rigidity result. Also we deal with the relations between flat metrics and \(\sigma _2\)-curvature. With a suitable condition on the \(\sigma _2\)-curvature, we show that a metric has to be flat if it is close to a flat metric. We conclude this paper by proving that the three-dimensional torus does not admit a metric with constant scalar curvature and nonnegative \(\sigma _2\)-curvature unless it is flat. 相似文献
19.
Brian Clarke Yanir A. Rubinstein 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2013
We consider geometries on the space of Riemannian metrics conformally equivalent to the widely studied Ebin L2 metric. Among these we characterize a distinguished metric that can be regarded as a generalization of Calabi?s metric on the space of Kähler metrics to the space of Riemannian metrics, and we study its geometry in detail. Unlike the Ebin metric, its geodesic equation involves non-local terms, and we solve it explicitly by using a constant of the motion. We then determine its completion, which gives the first example of a metric on the space of Riemannian metrics whose completion is strictly smaller than that of the Ebin metric. 相似文献
20.
Dually flat Finsler metrics arise from information geometry which has attracted some geometers and statisticians. In this paper, we study dually flat general spherically symmetric Finsler metrics which are defined by a Euclidean metric and two related 1-forms. We give the equivalent conditions for those metrics to be locally dually flat. By solving the equivalent equations, a group of new locally dually flat Finsler metrics is constructed. 相似文献