Double field theory provides T-duality covariant generalized tensors that are natural extensions of the scalar and Ricci curvatures of Riemannian geometry. We search for a similar extension of the Riemann curvature tensor by developing a geometry based on the generalized metric and the dilaton. We find a duality covariant Riemann tensor whose contractions give the Ricci and scalar curvatures, b...

We consider deformations of metrics in a given conformal class such that the smallest eigenvalue of the Ricci tensor to be a constant. It is related to the notion of minimal volumes in comparison geometry. Such a metric with the smallest eigenvalue of the Ricci tensor to be a constant is an extremal metric of volume in a suitable sense in the conformal class. The problem is reduced to solve a P...

Abstract In this paper, we study $$\eta$$ ? -Ricci solitons on almost cosymplectic $$(k,\mu )$$ ( k , ? ) -manifolds. As an application, it is proved that if -metric with $$k&lt;0$$ &lt;</mml:mo...

Using Ernst's theory of complex potentials, a solution of the coupled Einstein-Maxwell equations in oblate spheroidal coordinates is obtained for a source possessing mass, electric charge, and angular momentum. The density of the electromagnetic energy is evaluated. Explicit expressions are derived for the components of the Ricci tensor Rmunu, and of the electromagnetic energy momentum tensor E...