Compact Space-like Hypersurfaces with Constant Scalar Curvature in Locally Symmetric Lorentz Spaces
نویسندگان
چکیده
Let Nn+p p be an (n + p)-dimensional connected semi-Riemannian manifold of index p. It is called a semi-definite space of index p. When we refer to index p, we mean that there are only p negative eigenvalues of semi-Riemannian metric of Nn+p p and the other eigenvalues are positive. In particular, Nn+1 1 is called a Lorentz space when p = 1. When the Lorentz space Nn+1 1 is of constant curvature c, we call it Lorentz space form, denote it by Nn+1 1 (c), with de Sitter space S n+1 1 (1) and anti-de Sitter space Hn+1 1 (−1) as its special cases. A hypersurface M of a Lorentz space is said to be space-like if the induced metric from that of the ambient space is positive definite. The authors in [3] introduced a class of Lorentz spaces M of index 1. Let ∇, K and R denote the semi-Riemannian connection, sectional curvature and curvature tensor on M , respectively. For constant c1, c2 and c3, they considered Lorentz spaces which satisfy the following conditions: (1) for any space-like vector u and any time-like vector v, K(u, v) = − c1 n , (2) for any space-like vector u and v, K(u, v) ≥ c2, (3) |∇R| ≤ c3 n .
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