Abstract
Solutions to vacuum Einstein field equations with cosmological constant, such as the de Sitter space and the anti-de Sitter space, are basic in different cosmological and theoretical developments. It is also well known that complex structures admit metrics of this type. The most famous example is the complex projective space endowed with the Fubini-Study metric. In this work, we perform a systematic study of Einstein complex geometries derived from a logarithmic K\"ahler potential. Depending on the different contribution to the argument of such logarithmic term, we shall distinguish among direct, inverted and hybrid coordinates. They are directly related to the signature of the metric and determine the maximum domain of the complex space where the geometry can be defined.
Highlights
Complex manifolds have been implemented in modern theories, mainly within string frameworks
Two-dimensional differentiable manifolds with transition functions being conformal are isomorphic to complex manifolds of dimension one (Riemann surfaces), so the theory can be formulated on a one-dimensional complex manifold
Consider a complex manifold M and a Hermitian metric g on it; the necessary and sufficient condition for the metric g to be a Kähler one is ∇lcJ 1⁄4 0 [7], where J is the complex structure and ∇lc is the Levi-Civita connection
Summary
Complex manifolds have been implemented in modern theories, mainly within string frameworks. We are interested in the implementation of the concepts of complex differential geometry on extensible (not compactified) space-time structures. Within real manifolds, it is well known the existence of solutions of the vacuum Einstein field equations with cosmological constant. In the case of positive-definite signatures and constant bisectional curvature, we can distinguish among the complex projective space with the Fubini-Study (FS) metric (which has positive curvature), the Euclidean complex space (flat), and the unit ball (negative curvature).
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