共查询到20条相似文献,搜索用时 15 毫秒
1.
Alireza Ranjbar-Motlagh 《Differential Geometry and its Applications》2008,26(3):339-345
The purpose of this paper is to generalize the Liouville theorem for functions which are defined on the complete Riemannian manifolds. Then, we apply it to the isometric immersions between complete Riemannian manifolds in order to obtain an estimate for the size of the image of immersions in terms of the supremum of the length of their mean curvature vector in a quite general setting. The proofs are based on the Calabi's generalization of maximum principle for functions which are not necessarily differentiable. 相似文献
2.
We establish some perturbed minimization principles, and we develop a theory of subdifferential calculus, for functions defined on Riemannian manifolds. Then we apply these results to show existence and uniqueness of viscosity solutions to Hamilton–Jacobi equations defined on Riemannian manifolds. 相似文献
3.
João Xavier da Cruz Neto Ítalo Dowell Lira Melo Paulo Alexandre Araújo Sousa 《Journal of Optimization Theory and Applications》2017,173(2):459-470
The main goal of this paper is to present results of existence and nonexistence of convex functions on Riemannian manifolds, and in the case of the existence, we associate such functions to the geometry of the manifold. Precisely, we prove that the conservativity of the geodesic flow on a Riemannian manifold with infinite volume is an obstruction to the existence of convex functions. Next, we present a geometric condition that ensures the existence of (strictly) convex functions on a particular class of complete manifolds, and we use this fact to construct a manifold whose sectional curvature assumes any real value greater than a negative constant and admits a strictly convex function. In the last result, we relate the geometry of a Riemannian manifold of positive sectional curvature with the set of minimum points of a convex function defined on the manifold. 相似文献
4.
Orizon P. Ferreira 《Journal of Mathematical Analysis and Applications》2006,313(2):587-597
A characterization of Lipschitz behavior of functions defined on Riemannian manifolds is given in this paper. First, it is extended the concept of proximal subgradient and some results of proximal analysis from Hilbert space to Riemannian manifold setting. A technique introduced by Clarke, Stern and Wolenski [F.H. Clarke, R.J. Stern, P.R. Wolenski, Subgradient criteria for monotonicity, the Lipschitz condition, and convexity, Canad. J. Math. 45 (1993) 1167-1183], for generating proximal subgradients of functions defined on a Hilbert spaces, is also extended to Riemannian manifolds in order to provide that characterization. A number of examples of Lipschitz functions are presented so as to show that the Lipschitz behavior of functions defined on Riemannian manifolds depends on the Riemannian metric. 相似文献
5.
S. Hosseini M.R. Pouryayevali 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(12):3884-3895
In this paper, a notion of generalized gradient on Riemannian manifolds is considered and a subdifferential calculus related to this subdifferential is presented. A characterization of the tangent cone to a nonempty subset S of a Riemannian manifold M at a point x is obtained. Then, these results are applied to characterize epi-Lipschitz subsets of complete Riemannian manifolds. 相似文献
6.
J.-P. Ezin 《Advances in Mathematics》2002,172(2):206-224
Given a real number ε>0, small enough, an associated Jost map Jε between two Riemannian manifolds is defined. Then we prove that connected Riemannian manifolds for which the center of mass of each small geodesic ball is the center of the ball (i.e. for which the identity is a Jε map) are ball-homogeneous. In the analytic case we characterize such manifolds in terms of the Euclidean Laplacian and we show that they have constant scalar curvature. Under some restriction on the Ricci curvature we prove that Riemannian analytic manifolds for which the center of mass of each small geodesic ball is the center of the ball are locally and weakly harmonic. 相似文献
7.
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
8.
Kwang-Soon Park 《Quaestiones Mathematicae》2018,41(1):1-14
As a generalization of slant submersions [18], semi-slant submersions [15], and slant Riemannian maps [21], we define the notion of semi-slant Riemannian maps from almost Hermitian manifolds to Riemannian manifolds. We study the integrability of distributions, the geometry of fibers, the harmonicity of such maps, etc. We also find a condition for such maps to be totally geodesic and investigate some decomposition theorems. Moreover, we give examples. 相似文献
9.
Bayram Ṣahin 《Indagationes Mathematicae》2012,23(1-2):80-94
We construct Gauss–Weingarten-like formulas and define O’Neill’s tensors for Riemannian maps between Riemannian manifolds. By using these new formulas, we obtain necessary and sufficient conditions for Riemannian maps to be totally geodesic. Then we introduce semi-invariant Riemannian maps from almost Hermitian manifolds to Riemannian manifolds, give examples and investigate the geometry of leaves of the distributions defined by such maps. We also obtain necessary and sufficient conditions for semi-invariant maps to be totally geodesic and find decomposition theorems for the total manifold. Finally, we give a classification result for semi-invariant Riemannian maps with totally umbilical fibers. 相似文献
10.
A. Asanjarani 《Differential Geometry and its Applications》2008,26(4):434-444
By using a certain second order differential equation, the notion of adapted coordinates on Finsler manifolds is defined and some classifications of complete Finsler manifolds are found. Some examples of Finsler metrics, with positive constant sectional curvature, not necessarily of Randers type nor projectively flat, are found. This work generalizes some results in Riemannian geometry and open up, a vast area of research on Finsler geometry. 相似文献
11.
We study minimal graphic functions on complete Riemannian manifolds ∑ with nonnegative Ricci curvature, euclidean volume growth, and quadratic curvature decay. We derive global bounds for the gradients for minimal graphic functions of linear growth only on one side. Then we can obtain a Liouville‐type theorem with such growth via splitting for tangent cones of ∑ at infinity. When, in contrast, we do not impose any growth restrictions for minimal graphic functions, we also obtain a Liouville‐type theorem under a certain nonradial Ricci curvature decay condition on ∑. In particular, the borderline for the Ricci curvature decay is sharp by our example in the last section. © 2015 Wiley Periodicals, Inc. 相似文献
12.
There is a well-developed theory of weakly symmetric Riemannian manifolds. Here it is shown that several results in the Riemannian
case are also valid for weakly symmetric pseudo-Riemannian manifolds, but some require additional hypotheses. The topics discussed
are homogeneity, geodesic completeness, the geodesic orbit property, weak symmetries, and the structure of the nilradical
of the isometry group. Also, we give a number of examples of weakly symmetric pseudo-Riemannian manifolds, some mirroring
the Riemannian case and some indicating the problems in extending Riemannian results to weakly symmetric pseudo-Riemannian
spaces. 相似文献
13.
Vanishing theorems for harmonic and infinitesimal harmonic transformations of complete Riemannian manifolds are proved. The proof uses well-known Liouville theorems on subharmonic functions on noncompact complete Riemannian manifolds. 相似文献
14.
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
15.
Manfredo P. do Carmo Qiaoling Wang Changyu Xia 《Annali di Matematica Pura ed Applicata》2010,189(4):643-660
In this paper, we study eigenvalues of elliptic operators in divergence form on compact Riemannian manifolds with boundary
(possibly empty) and obtain a general inequality for them. By using this inequality, we prove universal inequalities for eigenvalues
of elliptic operators in divergence form on compact domains of complete submanifolds in a Euclidean space, and of complete
manifolds admitting special functions which include the Hadamard manifolds with Ricci curvature bounded below, a class of
warped product manifolds, the product of Euclidean spaces with any complete manifold and manifolds admitting eigenmaps to
a sphere. 相似文献
16.
In this paper, we generalize geodesic $E$-convex function and define geodesic $\gamma$-pre-$E$-convex and geodesic $\gamma$-$E$-convex functions on Riemannian manifolds. The sufficient condition of equivalence class of geodesic $\gamma$-pre-$E$-convexity and geodesic $\gamma$-$E$-convexity for differentiable function on Riemannian manifolds is studied. We discuss the sufficient condition for $E$-epigraph to be geodesic $E$-convex set. At the end, we establish some optimality results with the aid of geodesic $\gamma$-pre-$E$-convex and geodesic $\gamma$-$E$-convex functions and discuss the mean value inequality for geodesic $\gamma$-pre-$E$-convex function. 相似文献
17.
Atsushi Tachikawa 《manuscripta mathematica》1983,42(1):11-40
It is well known that the weakly harmonic mapping U∶M→N (M,N: Riemannian manifolds) is regular if the image U(M) is contained in some sufficiently small ball and for this case Liouville's theorem is valid. In this paper we show that the smallness condition for U(M) can be released if U minimizes the energy functional and the sectional curvatures of the target manifold N are bounded by some suitable function of the distance from some fixed point of N. 相似文献
18.
Following Mark Kac, it is said that a geometric property of a compact Riemannian manifold can be heard if it can be determined from the eigenvalue spectrum of the associated Laplace operator on functions. On the contrary, D’Atri
spaces, manifolds of type A{\mathcal{A}}, probabilistic commutative spaces,
\mathfrakC{\mathfrak{C}}-spaces,
\mathfrakTC{\mathfrak{TC}}-spaces, and
\mathfrakGC{\mathfrak{GC}}-spaces have been studied by many authors as symmetric-like Riemannian manifolds. In this article, we prove that for closed
Riemannian manifolds, none of the properties just mentioned can be heard. Another class of interest is the class of weakly
symmetric manifolds. We consider the local version of this property and show that weak local symmetry is another inaudible
property of Riemannian manifolds. 相似文献
19.
A Lie algebra g is called two step nilpotent if g is not abelian and [g, g] lies in the center of g. Two step nilpotent Lie algebras are useful in the study of some geometric problems, such as commutative Riemannian manifolds, weakly symmetric Riemannian manifolds, homogeneous Einstein manifolds, etc. Moreover, the classification of two-step nilpotent Lie algebras has been an important problem in Lie theory. In this paper, we study two step nilpotent indecomposable Lie algebras of dimension 8 over the field of complex numbers. Based on the study of minimal systems of generators, we choose an appropriate basis and give a complete classification of two step nilpotent Lie algebras of dimension 8. 相似文献
20.
We introduce the notion of geometrical engagement for actions of semisimple Lie groups and their lattices as a concept closely
related to Zimmer's topological engagement condition. Our notion is a geometrical criterion in the sense that it makes use
of Riemannian distances. However, it can be used together with the foliated harmonic map techniques introduced in [8] to establish
foliated geometric superrigidity results for both actions and geometric objects. In particular, we improve the applications
of the main theorem in [9] to consider nonpositively curved compact manifolds (not necessarily with strictly negative curvature).
We also establish topological restrictions for Riemannian manifolds whose universal cover have a suitable symmetric de Rham
factor (Theorem B), as well as geometric obstructions for nonpositively curved compact manifolds to have fundamental groups
isomorphic to certain groups build out of cocompact lattices in higher rank simple Lie groups (Corollary 4.5).
Received: October 22, 1997 相似文献