scholarly journals Mixed sectional-Ricci curvature obstructions on tori

2018 ◽  
Vol 12 (03) ◽  
pp. 713-734
Author(s):  
Benoît Kloeckner ◽  
Stéphane Sabourau
Keyword(s):  
Sectional Curvature ◽  
Ricci Curvature ◽  
Positive Constant ◽  
Riemannian Metrics ◽  
Small Positive Constant ◽  
Sectional And Ricci Curvatures

We establish new obstruction results to the existence of Riemannian metrics on tori satisfying mixed bounds on both their sectional and Ricci curvatures. More precisely, from Lohkamp’s theorem, every torus of dimension at least three admits Riemannian metrics with negative Ricci curvature. We show that the sectional curvature of these metrics cannot be bounded from above by an arbitrarily small positive constant. In particular, if the Ricci curvature of a Riemannian torus is negative, bounded away from zero, then there exist some planar directions in this torus where the sectional curvature is positive, bounded away from zero. All constants are explicit and depend only on the dimension of the torus.

scholarly journals A note on complete hypersurfaces of non-positive Ricci curvature

1983 ◽  
Vol 28 (3) ◽  
pp. 339-342 ◽  
Author(s):  
G.H. Smith
Keyword(s):  
Euclidean Space ◽  
Recent Result ◽  
Sectional Curvature ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Positive Constant ◽  
Constant Sectional Curvature ◽  
Positive Ricci Curvature ◽  
Simply Connected ◽  
Complete Hypersurfaces

In this note we point out that a recent result of Leung concerning hypersurfaces of a Euclidean space has a simple generalisation to hypersurfaces of complete simply-connected Riemannian manifolds of non-positive constant sectional curvature.

scholarly journals Smoothing Riemannian metrics with Ricci curvature bounds

1996 ◽  
Vol 90 (1) ◽  
pp. 49-61 ◽  
Author(s):  
Xianzhe Dai ◽  
Guofang Wei ◽  
Rugang Ye
Keyword(s):  
Ricci Curvature ◽  
Riemannian Metrics ◽  
Curvature Bounds

Uniform asymptotic expansions for oblate spheroidal functions I: positive separation parameter λ

1992 ◽  
Vol 121 (3-4) ◽  
pp. 303-320 ◽  
Author(s):  
T. M. Dunster
Keyword(s):  
Wave Equation ◽  
Half Plane ◽  
Positive Constant ◽  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Small Positive Constant ◽  
Separation Parameter ◽  
Oblate Spheroidal ◽  
Uniform Asymptotic Expansions

SynopsisUniform asymptotic expansions are derived for solutions of the spheroidal wave equation, in the oblate case where the parameter µ is real and nonnegative, the separation parameter λ is real and positive, and γ is purely imaginary (γ = iu). As u →∞, three types of expansions are derived for oblate spheroidal functions, which involve elementary, Airy and Bessel functions. Let δ be an arbitrary small positive constant. The expansions are uniformly valid for λ/u2 fixed and lying in the interval (0,2), and for λ / u2when 0<λ/u2 < 1, and when 1 = 1≦λ/u2 < 2. The union of the domains of validity of the various expansions cover the half- plane arg (z)≦ = π/2.

scholarly journals Small covers, infra-solvmanifolds and curvature

2015 ◽  
Vol 27 (5) ◽  
Author(s):  
Shintarô Kuroki ◽  
Mikiya Masuda ◽  
Li Yu
Keyword(s):  
Sectional Curvature ◽  
Riemannian Metrics ◽  
Flat Torus ◽  
Combinatorial Structures ◽  
Simple Polytope ◽  
Simple Polytopes ◽  
Small Cover ◽  
Equivalent Conditions

AbstractIt is shown that a small cover (resp. real moment-angle manifold) over a simple polytope is an infra-solvmanifold if and only if it is diffeomorphic to a real Bott manifold (resp. flat torus). Moreover, we obtain several equivalent conditions for a small cover to be homeomorphic to a real Bott manifold. In addition, we study Riemannian metrics on small covers and real moment-angle manifolds with certain conditions on the Ricci or sectional curvature. We will see that these curvature conditions put very strong restrictions on the topology of the corresponding small covers and real moment-angle manifolds and the combinatorial structures of the underlying simple polytopes.

Homogeneous Einstein Finsler Metrics on -dimensional Spheres

2018 ◽  
Vol 62 (3) ◽  
pp. 509-523
Author(s):  
Libing Huang ◽  
Xiaohuan Mo
Keyword(s):  
Sectional Curvature ◽  
Ricci Curvature ◽  
Einstein Metrics ◽  
Constant Sectional Curvature ◽  
Einstein Metric ◽  
Finsler Metrics ◽  
Dimensional Sphere ◽  
Order Ordinary Differential Equation ◽  
Canonical Metric ◽  
Homogeneous Einstein Metrics

AbstractIn this paper, we study a class of homogeneous Finsler metrics of vanishing $S$-curvature on a $(4n+3)$-dimensional sphere. We find a second order ordinary differential equation that characterizes Einstein metrics with constant Ricci curvature $1$ in this class. Using this equation we show that there are infinitely many homogeneous Einstein metrics on $S^{4n+3}$ of constant Ricci curvature $1$ and vanishing $S$-curvature. They contain the canonical metric on $S^{4n+3}$ of constant sectional curvature $1$ and the Einstein metric of non-constant sectional curvature given by Jensen in 1973.

Mean Curvature Comparison with L1-norms of Ricci Curvature

2004 ◽  
Vol 47 (2) ◽  
pp. 314-320 ◽  
Author(s):  
Jong-Gug Yun
Keyword(s):  
Ricci Curvature ◽  
Comparison Theorem ◽  
Positive Constant ◽  
Mean Curvature

AbstractWe prove an analogue of mean curvature comparison theorem in the case where the Ricci curvature below a positive constant is small in L1-norm.

scholarly journals Rigidity of volume-minimising hypersurfaces in Riemannian 5-manifolds

2018 ◽  
Vol 167 (02) ◽  
pp. 345-353
Author(s):  
ABRAÃO MENDES
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Positive Constant ◽  
Homotopy Class ◽  
Upper Bound ◽  
Rigidity Result ◽  
Nonnegative Ricci Curvature ◽  
Closed Hypersurface

AbstractIn this paper we generalise the main result of [4] for manifolds that are not necessarily Einstein. In fact, we obtain an upper bound for the volume of a locally volume-minimising closed hypersurface Σ of a Riemannian 5-manifold M with scalar curvature bounded from below by a positive constant in terms of the total traceless Ricci curvature of Σ. Furthermore, if Σ saturates the respective upper bound and M has nonnegative Ricci curvature, then Σ is isometric to 𝕊4 up to scaling and M splits in a neighbourhood of Σ. Also, we obtain a rigidity result for the Riemannian cover of M when Σ minimises the volume in its homotopy class and saturates the upper bound.

scholarly journals Biharmonic submanifolds in manifolds with bounded curvature

2016 ◽  
Vol 27 (11) ◽  
pp. 1650089
Author(s):  
Shun Maeta
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Ricci Curvature ◽  
Mean Curvature ◽  
Biharmonic Submanifolds ◽  
Bounded Curvature ◽  
Negative Constant ◽  
The Mean ◽  
Biharmonic Submanifold

We consider a complete biharmonic submanifold [Formula: see text] in a Riemannian manifold with sectional curvature bounded from above by a non-negative constant [Formula: see text]. Assume that the mean curvature is bounded from below by [Formula: see text]. If (i) [Formula: see text], for some [Formula: see text], or (ii) the Ricci curvature of [Formula: see text] is bounded from below, then the mean curvature is [Formula: see text]. Furthermore, if [Formula: see text] is compact, then we obtain the same result without the assumption (i) or (ii). These are affirmative partial answers to Balmuş–Montaldo–Oniciuc conjecture.

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