共查询到20条相似文献,搜索用时 78 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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3.
Tomasz Rybicki 《Geometriae Dedicata》2011,151(1):175-186
It is shown that certain diffeomorphism or homeomorphism groups with no restriction on support of an open manifold (being
the interior of a compact manifold) are bounded. It follows that these groups are uniformly perfect. In order to characterize
the boundedness several conditions on automorphism groups of an open manifold are introduced. In particular, it is shown that
the commutator length diameter of the automorphism group D(M){\mathcal{D}(M)} of a portable manifold M is estimated by 4. 相似文献
4.
Anton A. Ayzenberg 《Proceedings of the Steklov Institute of Mathematics》2018,302(1):16-32
We consider an effective action of a compact (n ? 1)-torus on a smooth 2n-manifold with isolated fixed points. We prove that under certain conditions the orbit space is a closed topological manifold. In particular, this holds for certain torus actions with disconnected stabilizers. There is a filtration of the orbit manifold by orbit dimensions. The subset of orbits of dimensions less than n ? 1 has a specific topology, which is axiomatized in the notion of a sponge. In many cases the original manifold can be recovered from its orbit manifold, the sponge, and the weights of tangent representations at fixed points. We elaborate on the introduced notions using specific examples: the Grassmann manifold G4,2, the complete flag manifold F3, and quasitoric manifolds with an induced action of a subtorus of complexity 1. 相似文献
5.
Irene Paniccia 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(3):255-271
The aim of this article is to prove a global existence result with small data for the heat flow for harmonic maps from a manifold
flat at infinity into a compact manifold. By flat at infinity we mean that the growth rate of the volumes of the balls on
the manifold is the same as in the flat space. This is true for any manifold for small enough radius, but is in general not
true when the radius of the ball grows. So prescribing such a growth rate also at infinity selects a class of manifolds on
which our result holds. In this setting estimates are available for the heat kernel and its gradient on the base manifold.
From such estimates it is easy to get L
p
−L
q
bounds for the heat kernel. A contraction principle argument then yields a local existence result in a suitable Sobolev space
and a global existence result for small data. 相似文献
6.
Hirokazu Shimobe 《Geometriae Dedicata》2018,197(1):49-60
We study compact complex manifolds bimeromorphic to locally conformally Kähler (LCK) manifolds. This is an analogy of studying a compact complex manifold bimeromorphic to a Kähler manifold. We give a negative answer for a question of Ornea, Verbitsky, Vuletescu by showing that there exists no LCK current on blow ups along a submanifold (dim \(\ge 1\)) of Vaisman manifolds. We show that a compact complex manifold with LCK currents satisfying a certain condition can be modified to an LCK manifold. Based on this fact, we define a compact complex manifold with a modification from an LCK manifold as a locally conformally class C (LC class C) manifold. We give examples of LC class C manifolds that are not LCK manifolds. Finally, we show that all LC class C manifolds are locally conformally balanced manifolds. 相似文献
7.
Vladimir S. Matveev 《Inventiones Mathematicae》2003,151(3):579-609
We show that if all geodesics of two non-proportional metrics on a closed manifold coincide (as unparameterized curves), then
the manifold has a finite fundamental group or admits a local-product structure. This implies that, if the manifold admits
a metric of negative sectional curvature, then two metrics on the manifold have the same geodesics if and only if they are
proportional.
Oblatum 18-IV-2002 & 12-VIII-2002?Published online: 18 December 2002 相似文献
8.
S. Deshmukh 《Annali dell'Universita di Ferrara》2011,57(1):17-26
It is known that a conformal vector field on a compact Kaehler manifold is a Killing vector field. In this paper, we are interested
in finding conditions under which a conformal vector field on a non-compact Kaehler manifold is Killing. First we prove that
a harmonic analytic conformal vector field on a 2n-dimensional Kaehler manifold (n ≠ 2) of constant nonzero scalar curvature is Killing. It is also shown that on a 2n-dimensional Kaehler Einstein manifold (n > 1) an analytic conformal vector field is either Killing or else the Kaehler manifold is Ricci flat. In particular, it follows
that on non-flat Kaehler Einstein manifolds of dimension greater than two, analytic conformal vector fields are Killing. 相似文献
9.
A hypercomplex structure on a differentiable manifold consists of three integrable almost complex structures that satisfy quaternionic relations. If, in addition, there exists a metric on the manifold which is Hermitian with respect to the three structures, and such that the corresponding Hermitian forms are closed, the manifold is said to be hyperkähler. In the paper “Non-Hermitian Yang–Mills connections” [13], Kaledin and Verbitsky proved that the twistor space of a hyperkähler manifold admits a balanced metric; these were first studied in the article “On the existence of special metrics in complex geometry” [17] by Michelsohn. In the present article, we review the proof of this result and then generalize it and show that twistor spaces of general compact hypercomplex manifolds are balanced. 相似文献
10.
In this paper, we introduce horizontal and vertical warped product Finsler manifolds. We prove that every C-reducible or proper Berwaldian doubly warped product Finsler manifold is Riemannian. Then, we find the relation between Riemannian curvatures of doubly warped product Finsler manifold and its components, and consider the cases that this manifold is flat or has scalar flag curvature. We define the doubly warped Sasaki-Matsumoto metric for warped product manifolds and find a condition under which the horizontal and vertical tangent bundles are totally geodesic. We obtain some conditions under which a foliated manifold reduces to a Reinhart manifold. Finally, we study an almost complex structure on the tangent bundle of a doubly warped product Finsler manifold. 相似文献
11.
A. Arouche M. Deffaf A. Zeghib 《Transactions of the American Mathematical Society》2007,359(3):1253-1263
We show a geometric rigidity of isometric actions of non-compact (semisimple) Lie groups on Lorentz manifolds. Namely, we show that the manifold has a warped product structure of a Lorentz manifold with constant curvature by a Riemannian manifold.
12.
M. Cencelj Yu. V. Muranov D. Repovš 《Abhandlungen aus dem Mathematischen Seminar der Universit?t Hamburg》2006,76(1):35-55
The problem of splitting a homotopy equivalence along a submanifold is closely related to the surgery exact sequence and to
the problem of surgery of manifold pairs. In classical surgery theory there exist two approaches to surgery in the category
of manifolds with boundaries. In the rel ∂ case the surgery on a manifold pair is considered with the given fixed manifold
structure on the boundary. In the relative case the surgery on the manifold with boundary is considered without fixing maps
on the boundary. Consider a normal map to a manifold pair (Y, ∂Y) ⊂ (X, ∂X) with boundary which is a simple homotopy equivalence on the boundary∂X. This map defines a mixed structure on the manifold with the boundary in the sense of Wall. We introduce and study groups
of obstructions to splitting of such mixed structures along submanifold with boundary (Y, ∂Y). We describe relations of these groups to classical surgery and splitting obstruction groups. We also consider several geometric
examples. 相似文献
13.
In this article, we study curvatures on a strongly convex (weakly) Kähler-Finsler manifold. First, we prove that the holomorphic sectional curvature is just half of the flag curvature in a holomorphic plane section on a strongly convex weakly Kähler-Finsler manifold. Second, we compare curvatures associated to the Rund connection with curvatures associated to the Chern-Finsler connection or the complex Berward connection on a strongly convex Kähler-Finsler manifold. Finally, we discuss relationships between flag curvatures and holomorphic bisectional curvatures, and compare two kinds of S-curvatures on a strongly convex Kähler-Finsler manifold. 相似文献
14.
We consider several transformation groups of a locally conformally Kähler manifold and discuss their inter-relations. Among other results, we prove that all conformal vector fields on a compact Vaisman manifold which is neither locally conformally hyperkähler nor a diagonal Hopf manifold are Killing, holomorphic and that all affine vector fields with respect to the minimal Weyl connection of a locally conformally Kähler manifold which is neither Weyl-reducible nor locally conformally hyperkähler are holomorphic and conformal. 相似文献
15.
Wilfried H. Paus 《Transactions of the American Mathematical Society》1998,350(10):3943-3966
In this paper, we investigate under what circumstances the Laplace-Beltrami operator on a pseudo-Riemannian manifold can be written as a sum of squares of vector fields, as is naturally the case in Euclidean space.
We show that such an expression exists globally on one-dimensional manifolds and can be found at least locally on any analytic pseudo-Riemannian manifold of dimension greater than two. For two-dimensional manifolds this is possible if and only if the manifold is flat.
These results are achieved by formulating the problem as an exterior differential system and applying the Cartan-Kähler theorem to it.
16.
Yiping Mao 《Proceedings of the American Mathematical Society》1997,125(9):2699-2702
In this short note we obtain a converse to the Gelfand theorem: a Riemannian manifold is homogeneous if the isometrically invariant operators on the manifold form a commutative algebra.
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We study the Lie algebra of infinitesimal isometries on compact Sasakian and K-contact manifolds. On a Sasakian manifold which is not a space form or 3-Sasakian, every Killing vector field is an infinitesimal automorphism of the Sasakian structure. For a manifold with K-contact structure, we prove that there exists a Killing vector field of constant length which is not an infinitesimal automorphism of the structure if and only if the manifold is obtained from the Konishi bundle of a compact pseudo-Riemannian quaternion-Kähler manifold after changing the sign of the metric on a maximal negative distribution. We also prove that nonregular Sasakian manifolds are not homogeneous and construct examples with cohomogeneity one. Using these results we obtain in the last section the classification of all homogeneous Sasakian manifolds. 相似文献
19.
W. -J. Beyn 《Numerical Functional Analysis & Optimization》2013,34(5-6):503-514
We consider systems of m nonlinear equations in m + p unknowns which have p-dimensional solution manifolds. It is well-known that the Gauss-Newton method converges locally and quadratically to regular points on this manifold. We investigate in detail the mapping which transfers the starting point to its limit on the manifold. This mapping is shown to be smooth of one order less than the given system. Moreover, we find that the Gauss-Newton method induces a foliation of the neighborhood of the manifold into smooth submanifolds. These submanifolds are of dimension m, they are invariant under the Gauss-Newton iteration, and they have orthogonal intersections with the solution manifold. 相似文献
20.
Brian Clarke 《Annals of Global Analysis and Geometry》2011,39(2):131-163
We study the manifold of all Riemannian metrics over a closed, finite-dimensional manifold. In particular, we investigate
the topology on the manifold of metrics induced by the distance function of the L
2 Riemannian metric—so-called because it induces an L
2 topology on each tangent space. It turns out that this topology on the tangent spaces gives rise to an L
1-type topology on the manifold of metrics itself. We study this new topology and its completion, which agrees homeomorphically
with the completion of the L
2 metric. We also give a user-friendly criterion for convergence (with respect to the L
2 metric) in the manifold of metrics. 相似文献