THE SPECTRAL GEOMETRY OF SOME ALMOST HERMITIAN MANIFOLDS

Abstract

Let $(M_i,g_i)$ be a certain almost Hermitian $2n$-manifold $M_i$ with a Hermitian metric $g_i$ for $i=1,2$, which is more general than an almost $L$ manifold (a Käshlerian manifold is known to be a special almost $L$ manifold). Let $Spe{c}^p(M_i,g_i)$ denote the spectrum of the real Laplacian on $p$-forms on $M_i$. The purpose of this paper is to show that for some special values of $p$ and $n$, if $Spe{c}^p(M_1,g_1)=Spe{c}^p(M_2,g_2)$, then $(M_1,g_1)$ is of constant holomorphic sectional curvature $H_1$ if and only if $(M_2,g_2)$ is of constant holomorphic sectional curvature $H_2$, and $H_2=H_1$. The corresponding results on almost $L$ manifolds were obtained by C. C. Hsiung and C. X. Wu (The spectral geometry of almost $L$ manifolds, Bull. Inst. Math. Acad. Sinica, 23 (1995), 229–241).