THE STRUCTURE OF THE REGULAR LEVEL SETS

Abstract

Consider the L 2 -adjoint s '� of the linearization of the scalar curvature s g. If ker s '� 0 on an n -dimensional compact manifold, it is well known that the scalar curvature s g is a non-negative constant. In this paper, we study the structure of the level set φ 1 (0) andnd the behavior of Ricci tensor when ker s '� 0 with s g > 0. Also for a non- trivial solution ( g; f ) of z = s '� ( f ) on an n -dimensional compact manifold, we analyze the structure of the regular level set f 1 ( 1). These results give a good understanding of the given manifolds.