LOCALLY CONFORMAL K¨AHLER MANIFOLDS AND CONFORMAL SCALAR CURVATURE
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초록
We show that on a compact locally conformal K¨ahler manifold M2n (dim M2n = 2n ≥ 4), M2n is Kahler if and only if its conformal scalar curvature k is not smaller than the scalar curvature s of M2n everywhere. As a consequence, if a compact locally conformal Kahler manifold M2n is both conformally flat and scalar flat, then M2n is Kahler. In contrast with the compact case, we show that there exists a locally conformal Kahler manifold with k equal to s, which is not Kahler.
Abstract
We show that on a compact locally conformal K¨ahler manifold M2n (dim M2n = 2n ≥ 4), M2n is Kahler if and only if its conformal scalar curvature k is not smaller than the scalar curvature s of M2n everywhere. As a consequence, if a compact locally conformal Kahler manifold M2n is both conformally flat and scalar flat, then M2n is Kahler. In contrast with the compact case, we show that there exists a locally conformal Kahler manifold with k equal to s, which is not Kahler.
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