scholarly journals Quantitative Rellich inequalities on Finsler—Hadamard manifolds

2016 ◽  
Vol 18 (06) ◽  
pp. 1650020 ◽  
Author(s):  
Alexandru Kristály ◽  
Dušan Repovš
Keyword(s):  
Riemannian Manifolds ◽  
Negative Curvature ◽  
Finsler Geometry ◽  
Flag Curvature ◽  
Hadamard Manifolds ◽  
Hardy Inequalities

In this paper, we are dealing with quantitative Rellich inequalities on Finsler–Hadamard manifolds where the remainder terms are expressed by means of the flag curvature. By exploring various arguments from Finsler geometry and PDEs on manifolds, we show that more weighty curvature implies more powerful improvements in Rellich inequalities. The sharpness of the involved constants is also studied. Our results complement those of Yang, Su and Kong [Hardy inequalities on Riemannian manifolds with negative curvature, Commun. Contemp. Math. 16 (2014), Article ID: 1350043, 24 pp.].

HARDY INEQUALITIES ON RIEMANNIAN MANIFOLDS WITH NEGATIVE CURVATURE

2014 ◽  
Vol 16 (02) ◽  
pp. 1350043 ◽  
Author(s):  
QIAOHUA YANG ◽  
DAN SU ◽  
YINYING KONG
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Negative Curvature ◽  
Geodesic Distance ◽  
Euclidean Spaces ◽  
Sharp Constants ◽  
Hardy Inequalities ◽  
Simply Connected

Let M be a complete, simply connected Riemannian manifold with negative curvature. We obtain the sharp constants of Hardy and Rellich inequalities related to the geodesic distance on M. Furthermore, if M is with strictly negative curvature, we show that the LpHardy inequalities can be globally refined by adding remainder terms like the Brezis–Vázquez improvement in case p ≥ 2, which is contrary to the case of Euclidean spaces.

scholarly journals Multipolar Hardy inequalities on Riemannian manifolds

2018 ◽  
Vol 24 (2) ◽  
pp. 551-567 ◽  
Author(s):  
Francesca Faraci ◽  
Csaba Farkas ◽  
Alexandru Kristály
Keyword(s):  
Variational Methods ◽  
Riemannian Manifolds ◽  
Quantitative Information ◽  
Beltrami Operator ◽  
Hadamard Manifolds ◽  
Hardy Inequalities ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Complete Riemannian Manifolds

We prove multipolar Hardy inequalities on complete Riemannian manifolds, providing various curved counterparts of some Euclidean multipolar inequalities due to Cazacu and Zuazua [Improved multipolar Hardy inequalities, 2013]. We notice that our inequalities deeply depend on the curvature, providing (quantitative) information about the deflection from the flat case. By using these inequalities together with variational methods and group-theoretical arguments, we also establish non-existence, existence and multiplicity results for certain Schrödinger-type problems involving the Laplace-Beltrami operator and bipolar potentials on Cartan-Hadamard manifolds and on the open upper hemisphere, respectively.

Invariant tensor fields of dynamical systems with pinched Lyapunov exponents and rigidity of geodesic flows

1989 ◽  
Vol 9 (3) ◽  
pp. 427-432 ◽  
Author(s):  
Renato Feres ◽  
Anatoly Katok
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Riemannian Manifolds ◽  
Negative Curvature ◽  
Geodesic Flows ◽  
Tensor Fields ◽  
Invariant Tensor ◽  
Affine Connections

AbstractWe consider in this note smooth dynamical systems equipped with smooth invariant affine connections and show that, under a pinching condition on the Lyapunov exponents, certain invariant tensor fields are parallel. We then apply this result to a problem of rigidity of geodesic flows for Riemannian manifolds with negative curvature.

scholarly journals A NEW QUANTITY IN RIEMANN-FINSLER GEOMETRY

2012 ◽  
Vol 54 (3) ◽  
pp. 637-645 ◽  
Author(s):  
XIAOHUAN MO ◽  
ZHONGMIN SHEN ◽  
HUAIFU LIU
Keyword(s):  
Scalar Curvature ◽  
Simple Proof ◽  
Finsler Geometry ◽  
Finsler Manifold ◽  
Flag Curvature ◽  
Finsler Metric ◽  
Finsler Metrics ◽  
Riemannian Curvature

AbstractIn this note, we study a new Finslerian quantity Ĉ defined by the Riemannian curvature. We prove that the new Finslerian quantity is a non-Riemannian quantity for a Finsler manifold with dimension n = 3. Then we study Finsler metrics of scalar curvature. We find that the Ĉ-curvature is closely related to the flag curvature and the H-curvature. We show that Ĉ-curvature gives, a measure of the failure of a Finsler metric to be of weakly isotropic flag curvature. We also give a simple proof of the Najafi-Shen-Tayebi' theorem.

scholarly journals A Kähler Einstein structure on the tangent bundle of a space form

2001 ◽  
Vol 25 (3) ◽  
pp. 183-195 ◽  
Author(s):  
Vasile Oproiu
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Tangent Bundle ◽  
Negative Curvature ◽  
Space Form ◽  
Positive Curvature ◽  
Holomorphic Sectional Curvature ◽  
Zero Section ◽  
Constant Negative Curvature ◽  
Constant Positive Curvature

We obtain a Kähler Einstein structure on the tangent bundle of a Riemannian manifold of constant negative curvature. Moreover, the holomorphic sectional curvature of this Kähler Einstein structure is constant. Similar results are obtained for a tube around zero section in the tangent bundle, in the case of the Riemannian manifolds of constant positive curvature.

scholarly journals Generalized Geodesic Convexity on Riemannian Manifolds

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 547 ◽  
Author(s):  
Izhar Ahmad ◽  
Meraj Ali Khan ◽  
Amira A. Ishan
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Global Minimum ◽  
Mean Value ◽  
Mathematical Programming Problem ◽  
Hadamard Manifolds ◽  
Geodesic Convexity ◽  
Invex Functions ◽  
Mean Value Inequality ◽  
Global Minimum Point

We introduce log-preinvex and log-invex functions on a Riemannian manifold. Some properties and relationships of these functions are discussed. A characterization for the existence of a global minimum point of a mathematical programming problem is presented. Moreover, a mean value inequality under geodesic log-preinvexity is extended to Cartan-Hadamard manifolds.

On Some Non-Riemannian Quantities in Finsler Geometry

2013 ◽  
Vol 56 (1) ◽  
pp. 184-193 ◽  
Author(s):  
Zhongmin Shen
Keyword(s):  
Finsler Geometry ◽  
Flag Curvature ◽  
Finsler Metrics ◽  
Geometric Properties

AbstractIn this paper we study several non-Riemannian quantities in Finsler geometry. These non- Riemannian quantities play an important role in understanding the geometric properties of Finsler metrics. In particular, we study a new non-Riemannian quantity defined by the S-curvature. We show some relationships among the flag curvature, the S-curvature, and the new non-Riemannian quantity.

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