Conformally flat 3-manifolds with constant scalar curvature II

First, we shall prove that a compact connected oriented locally conformally flat n-dimensional Riemannian manifold with constant scalar curvature is isometric to a space form or a Riemannian product Sn−1(c) × S1 if its Ricci curvature is nonnegative. Second, we shall give a topological classification of compact connected oriented locally conformally flat n-dimensional Riemannian manifolds with nonnegative scalar curvature r if the following inequality is satisfied: [sum ]i,jR2ij [les ] r2/(n−1), where [sum ]i,jR2ij is the squared norm of the Ricci curvature tensor.

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HYPERSURFACES IN A UNIT SPHERE Sn+1(1) WITH CONSTANT SCALAR CURVATURE

Journal of the London Mathematical Society ◽

10.1112/s0024610701002587 ◽

2001 ◽

Vol 64 (3) ◽

pp. 755-768 ◽

Cited By ~ 22

Author(s):

QING-MING CHENG

Keyword(s):

Scalar Curvature ◽

Unit Sphere ◽

Constant Scalar Curvature ◽

Second Fundamental Form ◽

Conformally Flat ◽

Principal Curvatures ◽

The Second Fundamental Form ◽

Locally Conformally Flat ◽

Complete Hypersurfaces ◽

Conformally Flat Hypersurface

The paper considers n-dimensional hypersurfaces with constant scalar curvature of a unit sphere Sn−1(1). The hypersurface Sk(c1)×Sn−k(c2) in a unit sphere Sn+1(1) is characterized, and it is shown that there exist many compact hypersurfaces with constant scalar curvature in a unit sphere Sn+1(1) which are not congruent to each other in it. In particular, it is proved that if M is an n-dimensional (n > 3) complete locally conformally flat hypersurface with constant scalar curvature n(n−1)r in a unit sphere Sn+1(1), then r > 1−2/n, and(1) when r ≠ (n−2)/(n−1), ifthen M is isometric to S1(√1−c2)×Sn−1(c), where S is the squared norm of the second fundamental form of M;(2) there are no complete hypersurfaces in Sn+1(1) with constant scalar curvature n(n−1)r and with two distinct principal curvatures, one of which is simple, such that r = (n−2)/(n−1) and

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Erratum to ``Conformally flat 3-manifolds with constant scalar curvature''

Journal of the Mathematical Society of Japan ◽

10.2969/jmsj/1191592002 ◽

2002 ◽

Vol 54 (4) ◽

pp. 997-1000

Author(s):

Qing-Ming CHENG ◽

Susumu ISHIKAWA ◽

Katsuhiro SHIOHAMA

Keyword(s):

Scalar Curvature ◽

Constant Scalar Curvature ◽

Conformally Flat

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Sufficient conditions for a pseudosymmetric spacetime to be a perfect fluid spacetime

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887821502170 ◽

2021 ◽

Author(s):

Peibiao Zhao ◽

Uday Chand De ◽

Bülent Ünal ◽

Krishnendu De

Keyword(s):

Cosmological Constant ◽

Scalar Curvature ◽

Perfect Fluid ◽

Sufficient Conditions ◽

Constant Scalar Curvature ◽

Conformally Flat ◽

Einstein’S Field Equations ◽

Field Equations ◽

Dust Fluid ◽

Perfect Fluid Spacetime

The aim of this paper is to obtain the condition under which a pseudosymmetric spacetime to be a perfect fluid spacetime. It is proven that a pseudosymmetric generalized Robertson–Walker spacetime is a perfect fluid spacetime. Moreover, we establish that a conformally flat pseudosymmetric spacetime is a generalized Robertson–Walker spacetime. Next, it is shown that a pseudosymmetric dust fluid with constant scalar curvature satisfying Einstein’s field equations without cosmological constant is vacuum. Finally, we construct a nontrivial example of pseudosymmetric spacetime.

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