Deformations and Einstein metrics I

This essay is about how to construct a new Einstein metric by an old one. Given an Einstein metric $α$ and its Killing $1$-form $β$, donote $b:=\|β\|_α$, we aim to determined the deformation factors $e^{ρ(b^2)}$ and $κ(b^2)$ such that $e^{ρ(b^2)}\sqrt{α^2-κ(b^2)β^2}$ 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 $β$ 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.