2-Type Surfaces of Constant Curvature in $S^n$

\S 0. Introduction.Let $M$ be a compact $C^{\infty}$ -Riemannian manifold, $C^{\infty}(M)$ the space of all smooth functions on $M$ , and $\Delta$ the Laplacian on $M$ .Then $\Delta$ is a self-adjoint elliptic differential operator acting on $C^{\infty}(M)$ , which has an infinite discrete sequence of eigenvalues:$Spec(M)=\{0=x_{0}<x_{1}<x_{2}<\cdots<x_{k}<\cdots\uparrow\infty\}$ .Let $V_{k}=V_{k}(M)$ be the eigenspace of $\Delta$ corresponding to the k-th eigenvalue $\lambda_{k}$ Then $V_{k}$ is finite-dimensional.We define an inner product $(, )$ on $C^{\infty}(M)$ by $(f, g)=\int_{M}fgdV$ , where $dV$ denotes the volume element on $M$ .Then $\sum_{t=0}^{\infty}V_{t}$ is dense in $C^{\infty}(M)$ and the decomposition is orthogonal with respect to the inner product $(, )$ .Thus we haveSince $M$ is compact, $V_{0}$ is the set of all constant functions which is 1- dimensional.Let $\tilde{M}$ be another compact $C^{\infty}$ -Riemannian manifold, and assume that $M$ is a submanifold of $\tilde{M}$ which is immersed by an isometric immersion $\varphi$ .We have the decomposition $C^{\infty}(\tilde{M})=\sum_{\iota=0}^{\infty}V_{\epsilon}(\tilde{M})$ (in $L^{2}$ -sense) with respect to the Laplacian $\Delta_{\tilde{M}}$ of $\tilde{M}$ .We denote by $\varphi^{*}$ the pull-back,