Deformations and Einstein metrics I

Abstract

This essay is about how to construct a new Einstein metric by an old one. Given an Einstein metric $\alpha$ and its Killing $1$-form $\beta$, donote $b:=\Vert \beta {\Vert }_{\alpha }$, we aim to determined the deformation factors $e^{\rho (b^2)}$ and $\kappa (b^2)$ such that $e^{\rho (b^2)}\sqrt{{\alpha }^2-\kappa (b^2){\beta }^2}$ becomes an Einstein metric. In face, it will depends critically on the peculiarities of the Killing $1$-form. As the first article in this series, we assume $\beta$ satisfies two curcial conditions (5.3) and (5.4), which are simple, natural and occursing only on even-dimensional manifolds. In this essay, we just need to regard the metric as a quadratic form. Any other additional structure on manifolds, such as topological structure, complex structure, etc., are not used.

arXiv comment27 pages,2 fugyre,1 table