Curvature bounds for the spectrum of a compact Riemannian manifold of constant scalar curvature

2005 ◽  
Vol 15 (4) ◽  
pp. 589-606 ◽  
Author(s):  
Sharief Deshmukh ◽  
Afifah Al-Eid
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Compact Riemannian Manifold ◽  
Constant Scalar Curvature ◽  
Curvature Bounds

EIGENVALUES OF GEOMETRIC OPERATORS RELATED TO THE WITTEN LAPLACIAN UNDER THE RICCI FLOW

2017 ◽  
Vol 59 (3) ◽  
pp. 743-751
Author(s):  
SHOUWEN FANG ◽  
FEI YANG ◽  
PENG ZHU
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Ricci Flow ◽  
Compact Riemannian Manifold ◽  
Laplacian Operator ◽  
Witten Laplacian ◽  
Normalized Ricci Flow ◽  
Geometric Operator

AbstractLet (M, g(t)) be a compact Riemannian manifold and the metric g(t) evolve by the Ricci flow. In the paper, we prove that the eigenvalues of geometric operator −Δφ + $\frac{R}{2}$ are non-decreasing under the Ricci flow for manifold M with some curvature conditions, where Δφ is the Witten Laplacian operator, φ ∈ C2(M), and R is the scalar curvature with respect to the metric g(t). We also derive the evolution of eigenvalues under the normalized Ricci flow. As a consequence, we show that compact steady Ricci breather with these curvature conditions must be trivial.

Curvature Bounds for the Spectrum of Closed Einstein Spaces

1978 ◽  
Vol 30 (5) ◽  
pp. 1087-1091 ◽  
Author(s):  
Udo Simon
Keyword(s):  
Lower Bound ◽  
Scalar Curvature ◽  
Sectional Curvature ◽  
Constant Scalar Curvature ◽  
Einstein Space ◽  
Curvature Bounds ◽  
Nonzero Eigenvalue ◽  
Einstein Spaces ◽  
Image Position

The following is our main result.(A) THEOREM. Let (M, g) be a closed connected Einstein space, n = dim M ≧ 2 (with constant scalar curvature R). Let K0 be the lower bound of the sectional curvature. Then either (M, g) is isometrically diffeomorphic to a sphere and the first nonzero eigenvalue ƛ1of the Laplacian fulfils

scholarly journals Torsional rigidity on compact Riemannian manifolds with lower Ricci curvature bounds

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Najoua Gamara ◽  
Abdelhalim Hasnaoui ◽  
Akrem Makni
Keyword(s):  
Riemannian Manifold ◽  
Dirichlet Problem ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Compact Riemannian Manifold ◽  
Torsional Rigidity ◽  
Hölder Inequality ◽  
Reverse Hölder Inequality ◽  
Curvature Bounds ◽  
Reverse Hölder

AbstractIn this article we prove a reverse Hölder inequality for the fundamental eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with lower Ricci curvature bounds. We also prove an isoperimetric inequality for the torsional ridigity of such domains

scholarly journals TRANSVERSAL INFINITESIMAL AUTOMORPHISMS ON KÄHLER FOLIATIONS

2012 ◽  
Vol 86 (3) ◽  
pp. 405-415 ◽  
Author(s):  
SEOUNG DAL JUNG ◽  
HUILI LIU
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Ricci Curvature ◽  
Compact Riemannian Manifold ◽  
Killing Field ◽  
Nonzero Constant ◽  
Conformal Field ◽  
Infinitesimal Automorphisms

AbstractLet ℱ be a Kähler foliation on a compact Riemannian manifold M. If the transversal scalar curvature of ℱ is nonzero constant, then any transversal conformal field is a transversal Killing field; and if the transversal Ricci curvature is nonnegative and positive at some point, then there are no transversally holomorphic fields.

scholarly journals Affine connections of non-integrable distributions

2020 ◽  
Vol 17 (08) ◽  
pp. 2050127
Author(s):  
Yong Wang
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Real Space ◽  
Space Forms ◽  
Affine Connections ◽  
Real Space Forms ◽  
Metric Connection ◽  
Statistical Connection

In this paper, we study non-integrable distributions in a Riemannian manifold with a semi-symmetric metric connection, a kind of semi-symmetric non-metric connection and a statistical connection. We obtain the Gauss, Codazzi, and Ricci equations for non-integrable distributions with respect to the semi-symmetric metric connection, the semi-symmetric non-metric connection and the statistical connection. As applications, we obtain Chen’s inequalities for non-integrable distributions of real space forms endowed with a semi-symmetric metric connection and a kind of semi-symmetric non-metric connection. We give some examples of non-integrable distributions in a Riemannian manifold with affine connections. We find some new examples of Einstein distributions and distributions with constant scalar curvature.

scholarly journals First Eigenvalues of Geometric Operator under The Ricci-Bourguignon Flow

2017 ◽  
Vol 24 (1) ◽  
pp. 51-60
Author(s):  
Shahroud Azami
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Compact Riemannian Manifold ◽  
Nonnegative Curvature ◽  
Laplacian Operator ◽  
Witten Laplacian ◽  
Geometric Operator

Let $(M,g(t))$ be a compact Riemannian manifold  and  the metric $g(t)$ evolve by the Ricci-Bourguignon flow. We find the formula variation of the eigenvalues of  geometric operator $-\Delta_{\phi}+cR$ under  the Ricci-Bourguignon flow, where  $\Delta_{\phi}$  is the Witten-Laplacian operator and $R$ is the scalar curvature. In the final  we show that some quantities dependent to the eigenvalues of  the geometric operator are  nondecreasing along the Ricci-Bourguignon flow on  closed manifolds  with nonnegative curvature.

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