Abstract
We investigate the spacelike hypersurface with constant scalar curvature (SCS) immersed in a Ricci symmetric manifold obeying standard curvature constraints. By supposing these hypersurfaces satisfy a suitable Okumura-type inequality recently introduced by Meléndez, which is a weaker hypothesis than to assume that they have two distinct principal curvatures, we obtain a series of umbilicity and pinching results. In particular, when the Ricci symmetric manifold is an Einstein manifold, then we further obtain some rigidity classifications of such hypersufaces.
Highlights
Let L1n+1 be an (n + 1)-dimensional Lorentz manifold, i.e., an indefinite Riemannian manifold of index 1
The problem of characterizing spacelike hypersurfaces immersed in a Lorentz space form is an important and fruitful topic in the theory of isometric immersions, which is originated from the seminal paper by Calabi in [1] and Cheng-Yau in [2]
As a generalization of their studies, it motivated a great deal of work of several authors to research the problem of hypersurface with constant mean curvature (CMC), such as [3,4,5], or constant scalar curvature (CSC), such as [6,7,8,9,10,11]
Summary
Let L1n+1 be an (n + 1)-dimensional Lorentz manifold, i.e., an indefinite Riemannian manifold of index 1. Are generalization of the Lorentz space forms and some non-trivial examples are given in [14,15,16] In this setting, many authors work in this type of ambient manifolds and a series of similar results for totally umbilical and pinching results are obtained (see [17,18,19]), but they could not give the rigidity classification results due to the fact that there are no nice symmetry properties for the ambient manifold. Ricci symmetric manifolds satisfying (1) and (2) which are not locally symmetric or space forms. In this sense, it is worth characterizing the spacelike hypersurfaces in such class of ambient manifolds
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