A curvature condition for a twisted product¶to be a warped product

Four-dimensional neutral signature self-dual gradient Ricci solitons

“…Since the onedimensional distribution span{∇f } is totally geodesic one has that (M, g) decomposes locally as a twisted product of the form I × ϕ N (see [24]). Moreover ρ(e 1 , e i ) = 0 (i = 2, 3, 4) shows that the twisted product reduces to a warped product [19]. Finally, since I × ϕ N is self-dual, it is necessarily locally conformally flat and the fiber N is of constant sectional curvature (see [6]).…”

“…Now, a straightforward calculation shows that the only possibly nonzero components of the Ricci tensor of g D,Φ are given by (19) ρ…”

“…On the other hand, since f is the potential function of a gradient Ricci soliton (i.e., Hes f +ρ = λg D,Φ ), it follows immediately from the expressions of the metric and the Ricci tensor in (18) and (19) that Hes f (∂ x i ′ , ∂ x j ′ ) = 0 for all i, j = 1, 2. Hence the function f expresses as…”

Reduced L_{q,p}-Cohomology of Some Twisted Products

“…Twisted products were the object of recent investigations [2,4,5,6,11,15]. The reduced L q,p -cohomology of warped cylinders [a, b) × h N , i.e., of product manifolds [a, b) × N endowed with a warped product metric…”

Special standard static space–times

“…There are also different generalizations of warped products such as warped products with more than one fiber manifold, called multiply warped products (see [35]) or warped products with two warping functions acting symmetrically on the fiber and base manifolds, called doubly warped products (see [34]). Finally, a warped product is said to be a twisted product if the warping function defined on the product of the base and fiber manifolds (see [16]). Basically, a standard static space-time can be considered as a Lorentzian warped product where the warping function is defined on a Riemannian manifold and acting on the negative definite metric on an open interval of real numbers.…”