In this article, we study the problem of existence and nonexistence warping function associated with constant scalar curvature on pseudo-Riemannian Poisson warped product space under assumption that fiber has curvature. We characterize Einstein by taking various dimensions base B (i.e; (1). \(dim B=1,\) (2). \(dimB\ge 2\)).

UST about a hundred years ago, Albert Einstein began writing a paper that secured his place in the pantheon of humankind's greatest thinkers. With his discovery of special relativity, Einstein upended the familiar, thousands-yearold conception of space and time. To be sure, even a century later, not everyone has fully embraced Einstein's discovery. Nevertheless, say "Einstein" and most everyone...

N. Hitchin [H] proved that the Euler characteristic χ(E) and signature σ(E) of a compact orientable 4-dimensional Einstein manifold E satisfy the inequality |σ(E)| 6 2 3χ(E), the equality holding only if either E is flat or the universal covering X of E is a K3-surface and π1(E) = 1, Z/2, or Z/2× Z/2. In the latter cases, E is a K3-surface if π1 = 1, an Enriques surface if π1 = Z/2, or the quot...

Using the new diffeomorphism invariants of Seiberg and Witten, a uniqueness theorem is proved for Einstein metrics on compact quotients of irreducible 4-dimensional symmetric spaces of non-compact type. The proof also yields a Riemannian version of the Miyaoka-Yau inequality. A smooth Riemannian manifold (M, g) is said [1] to be Einstein if its Ricci curvature is a constant multiple of g. Any i...

On any given compact manifold M with boundary ∂M , it is proved that the moduli space E of Einstein metrics on M , if non-empty, is a smooth, infinite dimensional Banach manifold, at least when π1(M,∂M) = 0. Thus, the Einstein moduli space is unobstructed. The usual Dirichlet and Neumann boundary maps to data on ∂M are smooth, but not Fredholm. Instead, one has natural mixed boundary-value prob...