On the Conformal Transformation Group of a Compact Riemannian Manifold with Constant Scalar Curvature

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.

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TRANSVERSAL INFINITESIMAL AUTOMORPHISMS ON KÄHLER FOLIATIONS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972711003431 ◽

2012 ◽

Vol 86 (3) ◽

pp. 405-415 ◽

Cited By ~ 3

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.

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The conformal transformation group of a compact homogeneous Riemannian manifold

Bulletin of the American Mathematical Society ◽

10.1090/s0002-9904-1962-10815-5 ◽

1962 ◽

Vol 68 (4) ◽

pp. 378-382 ◽

Cited By ~ 1

Author(s):

Samuel I. Goldberg ◽

Shoshichi Kobayashi

Keyword(s):

Riemannian Manifold ◽

Conformal Transformation ◽

Transformation Group ◽

Homogeneous Riemannian Manifold

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Affine connections of non-integrable distributions

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887820501273 ◽

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.

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