Antisymmetrised \(2p\)-forms generalising curvature 2-forms and a corresponding \(p\)-hierarchy of Schwarzschild type metrics in dimensions \(d>2p+1\) (Q5929514)

Generalizing the Einstein-Hilbert system in a quite natural fashion, \textit{A. Chakrabarti} and \textit{D. H. Tchrakian} [Phys. Lett., B 168, 187-191 (1986)] and \textit{G. M. O'Brian} and \textit{D. H. Tchrakian} [ J. Math. Phys. 29, 1212-1219 (1988; Zbl 0649.53082)] studied a hierarchy of gravitational systems. The prime objective of this paper is to present the Schwarzschild like solutions to such systems. The authors start with a curvature two-form and by recursive construction, introduce totally antisymmetric \(2p\)-forms, referred to as \(p\)-Riemann tensors which, on contraction of indices, gives a corresponding generalization of the Ricci tensor. In this context, the authors construct a static, spherically symmetric \(p\)-Ricci flat Schwarzschild like metric for spacetime dimension \(d> 2p+1\). It is worth noticing that \(2p\)-forms vanish for \(d<2p\) while the limiting cases \(d = 2p\) and \(d= 2p+1\) exhibit special features which are discussed briefly. Moreover, the existence of de Sitter type solutions is also pointed out. The authors then make an observation that for \(d=4p\), their class of solutions correspond to double-self dual \(p\)-Riemann tensors. Later on, the authors discuss the topological aspects of such generalized gravitational instantons and those of associated (through spin connections) generalized Yang-Mills instantons. In the concluding part of this paper, the authors make some pertinent comments on the possibility of a study of surface deformations at the horizons of their class of \(p\)-black holes leading to Virasoro algebras with a \(p\)-dependent hierarchy of central charges, and indicate directions for further study within the framework of the formulism of this paper in a broader context.