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 On a construction of complete simply-connected riemannian manifolds with negative curvature

1989 ◽  
Vol 113 ◽  
pp. 7-13
Author(s):  
Haruo Kitahara ◽  
Hajime Kawakami ◽  
Jin Suk Pak
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Negative Curvature ◽  
Harmonic Forms ◽  
Simply Connected

Let M be a complete simply-connected riemannian manifold of even dimension m. J. Dodziuk and I.M. Singer ([D1]) have conjectured that H2p(M) = 0 if p ≠ m/2 and dim H2m/2(M) = ∞, where H2*(M) is the space of L2-harmonic forms on M.

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 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.].

scholarly journals Geodesic flows of negatively curved manifolds with smooth stable and unstable foliations

1988 ◽  
Vol 8 (2) ◽  
pp. 215-239 ◽  
Author(s):  
Masahiko Kanai
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Geodesic Flows ◽  
Constant Negative Curvature ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Closed Riemannian Manifold

AbstractWe are concerned with closed C∞ riemannian manifolds of negative curvature whose geodesic flows have C∞ stable and unstable foliations. In particular, we show that the geodesic flow of such a manifold is isomorphic to that of a certain closed riemannian manifold of constant negative curvature if the dimension of the manifold is greater than two and if the sectional curvature lies between − and −1 strictly.

scholarly journals Warped Product Submanifolds in Locally Golden Riemannian Manifolds with a Slant Factor

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2125
Author(s):  
Cristina E. Hretcanu ◽  
Adara M. Blaga
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Warped Product ◽  
Euclidean Spaces ◽  
Slant Submanifolds

In the present paper, we study some properties of warped product pointwise semi-slant and hemi-slant submanifolds in Golden Riemannian manifolds, and we construct examples in Euclidean spaces. Additionally, we study some properties of proper warped product pointwise semi-slant (and, respectively, hemi-slant) submanifolds in a locally Golden Riemannian manifold.

scholarly journals Metric connections with parallel twistor-free torsion

2021 ◽  
pp. 2140011
Author(s):  
Andrei Moroianu ◽  
Mihaela Pilca
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Cyclic Component ◽  
Metric Connection ◽  
Simply Connected ◽  
Three Components

The torsion of every metric connection on a Riemannian manifold has three components: one totally skew-symmetric, one of vectorial type and one of twistorial type, which is also called the traceless cyclic component. In this paper we classify complete simply connected Riemannian manifolds carrying a metric connection whose torsion is parallel, has nonzero vectorial component and vanishing twistorial component.

Best Constants for Moser-Trudinger Inequalities on High Dimensional Hyperbolic Spaces

2013 ◽  
Vol 13 (4) ◽  
Author(s):  
Guozhen Lu ◽  
Hanli Tang
Keyword(s):  
Riemannian Manifolds ◽  
Best Constants ◽  
Hyperbolic Spaces ◽  
High Dimensional ◽  
Sharp Constant ◽  
Heisenberg Groups ◽  
Euclidean Spaces ◽  
Sharp Constants ◽  
Trudinger Inequality

AbstractThough there have been extensive works on best constants for Moser-Trudinger inequalities in Euclidean spaces, Heisenberg groups or compact Riemannian manifolds, much less is known for sharp constants for the Moser-Trudinger inequalities on hyperbolic spaces. Earlier works only include the sharp constant for the Moser-Trudinger inequality on the twodimensional hyperbolic disc. In this paper, we establish best constants for several types of Moser-Trudinger inequalities on high dimensional hyperbolic spaces ℍ

scholarly journals Non-reductive Homogeneous Pseudo-Riemannian Manifolds of Dimension Four

2006 ◽  
Vol 58 (2) ◽  
pp. 282-311 ◽  
Author(s):  
M. E. Fels ◽  
A. G. Renner
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Riemannian Manifolds ◽  
Homogeneous Spaces ◽  
Isotropy Subgroup ◽  
Algebraic Classification ◽  
Ricci Flat ◽  
Einstein Spaces ◽  
Simply Connected

AbstractA method, due to Élie Cartan, is used to give an algebraic classification of the non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Only one case with Lorentz signature can be Einstein without having constant curvature, and two cases with (2, 2) signature are Einstein of which one is Ricci-flat. If a four-dimensional non-reductive homogeneous pseudo-Riemannian manifold is simply connected, then it is shown to be diffeomorphic to ℝ4. All metrics for the simply connected non-reductive Einstein spaces are given explicitly. There are no non-reductive pseudo-Riemannian homogeneous spaces of dimension two and none of dimension three with connected isotropy subgroup.

scholarly journals On an Anti-Torqued Vector Field on Riemannian Manifolds

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2201
Author(s):  
Sharief Deshmukh ◽  
Ibrahim Al-Dayel ◽  
Devaraja Mallesha Naik
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Compact Riemannian Manifold ◽  
Killing Vector ◽  
Dual Vector ◽  
Definition Of ◽  
Simply Connected

A torqued vector field ξ is a torse-forming vector field on a Riemannian manifold that is orthogonal to the dual vector field of the 1-form in the definition of torse-forming vector field. In this paper, we introduce an anti-torqued vector field which is opposite to torqued vector field in the sense it is parallel to the dual vector field to the 1-form in the definition of torse-forming vector fields. It is interesting to note that anti-torqued vector fields do not reduce to concircular vector fields nor to Killing vector fields and thus, give a unique class among the classes of special vector fields on Riemannian manifolds. These vector fields do not exist on compact and simply connected Riemannian manifolds. We use anti-torqued vector fields to find two characterizations of Euclidean spaces. Furthermore, a characterization of an Einstein manifold is obtained using the combination of a torqued vector field and Fischer–Marsden equation. We also find a condition under which the scalar curvature of a compact Riemannian manifold admitting an anti-torqued vector field is strictly negative.

scholarly journals Some results about concircular vector fields on Riemannian manifolds

Filomat ◽  
2020 ◽  
Vol 34 (3) ◽  
pp. 835-842
Author(s):  
Bang-Yen Chen ◽  
Sharief Deshmukh
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Euclidean Spaces ◽  
Rigidity Results ◽  
Kaehler Manifolds

In this article, we show that the presence of a concircular vector field on a Riemannian manifold can be used to obtain rigidity results for Riemannian and Kaehler manifolds. More precisely, we find new geometrical characterizations of spheres, Euclidean spaces as well as of complex Euclidean spaces using non-trivial concircular vector fields.

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