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1.
In this paper nontrivial Killing vector fields of constant length and the corresponding ows on smooth complete Riemannian
manifolds are investigated. It is proved that such a ow on symmetric space is free or induced by a free isometric action of
the circle S
1. Examples of unit Killing vector fields generated by almost free but not free actions of S
1 on locally symmetric Riemannian spaces are found; among them are homogeneous (nonsimply connected) Riemannian manifolds of
constant positive sectional curvature and locally Euclidean spaces. Some unsolved questions are formulated.
DOI: . 相似文献
2.
We study the nontrivial Killing vector fields of constant length and the corresponding flows on smooth Riemannian manifolds. We describe the properties of the set of all points of finite (infinite) period for general isometric flows on Riemannian manifolds. It is shown that this flow is generated by an effective almost free isometric action of the group S 1 if there are no points of infinite or zero period. In the last case, the set of periods is at most countable and generates naturally an invariant stratification with closed totally geodesic strata; the union of all regular orbits is an open connected dense subset of full measure. 相似文献
3.
Killing vector fields of constant length correspond to isometries of constant displacement. Those in turn have been used to study homogeneity of Riemannian and Finsler quotient manifolds. Almost all of that work has been done for group manifolds or, more generally, for symmetric spaces. This paper extends the scope of research on constant length Killing vector fields to a class of Riemannian normal homogeneous spaces. 相似文献
4.
A vector field on Riemannian manifold is called conformal Killing if it generates oneparameter group of conformal transformation. The class of conformal Killing symmetric tensor fields of
an arbitrary rank is a natural generalization of the class of conformal Killing vector fields, and appears in different geometric
and physical problems. In this paper, we prove that a trace-free conformal Killing tensor field is identically zero if it
vanishes on some hypersurface. This statement is a basis of the theorem on decomposition of a symmetric tensor field on a
compact manifold with boundary to a sum of three fields of special types. We also establish triviality of the space of trace-free
conformal Killing tensor fields on some closed manifolds. 相似文献
5.
6.
Charles Frances 《Geometriae Dedicata》2012,158(1):35-59
We study pseudo-Riemannian conformal vector fields in the neighborhood of a singularity. For Riemannian manifolds, we prove that if a conformal vector field vanishing at a point x 0 is not Killing for a metric in the conformal class, then a neighborhood of the singularity x 0 is conformally flat. 相似文献
7.
A vector field X on a Riemannian manifold determines a submanifold in the tangent bundle. The volume of X is the volume of this submanifold for the induced Sasaki metric. When M is compact, the volume is well defined and, usually, this functional is studied for unit fields. Parallel vector fields are trivial minima of this functional.For manifolds of dimension 5, we obtain an explicit result showing how the topology of a vector field with constant length influences its volume. We apply this result to the case of vector fields that define Riemannian foliations with all leaves compact.Received: 29 April 2004 相似文献
8.
《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
9.
Yuri Bozhkov 《Applied Mathematics Letters》2010,23(10):1166-1169
We establish a generalization to Riemannian manifolds of the Caffarelli–Kohn–Nirenberg inequality. The applied method is based on the use of conformal Killing vector fields and E. Mitidieri’s approach to Hardy inequalities. 相似文献
10.
11.
Carlos Currás-Bosch 《Israel Journal of Mathematics》1986,53(3):315-320
In this paper we give a generalisation of Kostant’s Theorem about theA
x
-operator associated to a Killing vector fieldX on a compact Riemannian manifold. Kostant proved (see [6], [5] or [7]) that in a compact Riemannian manifold, the (1, 1)
skew-symmetric operatorA
x
=L
x
−≡
x
associated to a Killing vector fieldX lies in the holonomy algebra at each point. We prove that in a complete non-compact Riemannian manifold (M, g) theA
x
-operator associated to a Killing vector field, with finite global norm, lies in the holonomy algebra at each point. Finally
we give examples of Killing vector fields with infinite global norms on non-flat manifolds such thatA
x
does not lie in the holonomy algebra at any point. 相似文献
12.
S. E. Stepanov 《Theoretical and Mathematical Physics》2000,122(3):402-414
An analytic method, which Wu called the “Bochner technique,” has been used for fifty years to describe global Riemannian and Kähler geometries. We use this method to describe conformally Killing vector fields and harmonic timelike vector fields on a Lorentzian manifold and to study hydrodynamic models of the Universe, the existence of closed spacelike sections, and the possibility of fibering Lorentzian manifolds. 相似文献
13.
Cs. Vincze 《Differential Geometry and its Applications》2006,24(1):1-20
As it is well-known, a Minkowski space is a finite dimensional real vector space equipped with a Minkowski functional F. By the help of its second order partial derivatives we can introduce a Riemannian metric on the vector space and the indicatrix hypersurface S:=F−1(1) can be investigated as a Riemannian submanifold in the usual sense.Our aim is to study affine vector fields on the vector space which are, at the same time, affine with respect to the Funk metric associated with the indicatrix hypersurface. We give an upper bound for the dimension of their (real) Lie algebra and it is proved that equality holds if and only if the Minkowski space is Euclidean. Criteria of the existence is also given in lower dimensional cases. Note that in case of a Euclidean vector space the Funk metric reduces to the standard Cayley-Klein metric perturbed with a nonzero 1-form.As an application of our results we present the general solution of Matsumoto's problem on conformal equivalent Berwald and locally Minkowski manifolds. The reasoning is based on the theory of harmonic vector fields on the tangent spaces as Riemannian manifolds or, in an equivalent way, as Minkowski spaces. Our main result states that the conformal equivalence between two Berwald manifolds must be trivial unless the manifolds are Riemannian. 相似文献
14.
Gen-ichi Oshikiri 《manuscripta mathematica》2001,104(4):527-531
In this paper, we extend a result by H. Takagi on the non-existence of mutually commuting and linearly independent Killing
vector fields on positively curved Riemannian manifolds. Further, a kind of “Compact Leaf Theorem” is proved for metric foliations
of closed manifolds with positive sectional curvature.
Received: 26 May 2000 / Revised version: 28 February 2001 相似文献
15.
I. G. Nikolaev 《Commentarii Mathematici Helvetici》1995,70(1):210-234
Schur's theorem states that an isotropic Riemannian manifold of dimension greater than two has constant curvature. It is natural
to guess that compact almost isotropic Riemannian manifolds of dimension greater than two are close to spaces of almost constant
curvature. We take the curvature anisotropy as the discrepancy of the sectional curvatures at a point. The main result of
this paper is that Riemannian manifolds in Cheeger's class ℜ(n,d,V,A) withL
1-small integral anisotropy haveL
p-small change of the sectional curvature over the manifold. We also estimate the deviation of the metric tensor from that
of constant curvature in theW
p
2
-norm, and prove that compact almost isotropic spaces inherit the differential structure of a space form. These stability
results are based on the generalization of Schur' theorem to metric spaces. 相似文献
16.
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
17.
Bayram Sahin 《Proceedings Mathematical Sciences》2008,118(4):573-581
We study harmonic Riemannian maps on locally conformal Kaehler manifolds (lcK manifolds). We show that if a Riemannian holomorphic
map between lcK manifolds is harmonic, then the Lee vector field of the domain belongs to the kernel of the Riemannian map
under a condition. When the domain is Kaehler, we prove that a Riemannian holomorphic map is harmonic if and only if the lcK
manifold is Kaehler. Then we find similar results for Riemannian maps between lcK manifolds and Sasakian manifolds. Finally,
we check the constancy of some maps between almost complex (or almost contact) manifolds and almost product manifolds. 相似文献
18.
SHARIEF DESHMUKH 《Proceedings Mathematical Sciences》2011,121(2):171-179
In this paper, we classify real hypersurfaces in the complex projective space
C P\fracn+12C P^{\frac{n+1}{2}} whose structure vector field is a φ-analytic vector field (a notion similar to analytic vector fields on complex manifolds). We also define Jacobi-type vector
fields on a Riemannian manifold and classify real hypersurfaces whose structure vector field is a Jacobi-type vector field. 相似文献
19.
Philippe Tondeur 《Proceedings of the American Mathematical Society》1997,125(11):3403-3405
We prove that a flow on a closed manifold is Riemannian if and only if it is locally generated by Killing vector fields for a Riemannian metric.
20.
Domenico Perrone 《Acta Mathematica Hungarica》2013,138(1-2):102-126
We investigate almost contact metric manifolds whose Reeb vector field is a harmonic unit vector field, equivalently a harmonic section. We first consider an arbitrary Riemannian manifold and characterize the harmonicity of a unit vector field ??, when ??? is symmetric, in terms of Ricci curvature. Then, we show that for the class of locally conformal almost cosymplectic manifolds whose Reeb vector field ?? is geodesic, ?? is a harmonic section if and only if it is an eigenvector of the Ricci operator. Moreover, we build a large class of locally conformal almost cosymplectic manifolds whose Reeb vector field is a harmonic section. Finally, we exhibit several classes of almost contact metric manifolds where the associated almost contact metric structures ?? are harmonic sections, in the sense of Vergara-Diaz and Wood?[25], and in some cases they are also harmonic maps. 相似文献