Assume the Riemannian manifold \((M,\langle\;,\;\rangle)\) to have positive Ricci curvature; then by the Ricci tensor a second Riemannian metric, denoted by \(\langle\;,\;\rangle_ *\), is defined on \(M\). In the present paper the author studies this metric on ovaloids in Euclidean space \(E^{n+1}\). A first result is a correct version of an erroneous paper by \textit{T. Constantinescu} [Bull. Math. Soc. Sci. Math. Repub. Soc. Roum., Nouv. Ser. 25(73), 33-35 (1981; Zbl 0475.53050)]: Let \(M\) be an ovaloid in \(E^ 3\). Then \(K^*-1\) changes sign or \(M\) is a sphere and \(K^*=1\), where \(K^*\) denotes the Gauss curvature of the Ricci metric \(\langle\;,\;\rangle_ *\). The main part of the paper leads to the following generalization: Let \(M\) be an ovaloid in \(E^{n+1}\). Assume that the sectional curvature \(K^*\) of \(M\) with respect to the Ricci metric satisfies \(K^*\leq 1/(n-1)\); then \(M\) is a hypersphere and \(K^*=1/(n-1)\). At last the following theorem is proved: Let \(M\) be an ovaloid in \(E^{n+1}\). Then \(\max(\text{Ric}^*)\geq 1\), where \(\text{Ric}^*\) is the Ricci curvature of the Ricci metric. An important consequence of this theorem is the corollary: Let \((M,\bar g)\) be a compact, \(n\)-dimensional Riemannian manifold with Ricci curvature less than 1. Then there is no Riemannian metric \(g\), such that \((M,g)\) is realized as an ovaloid in \(E^{n+1}\) with \(\bar g\) as Ricci tensor. Remark: Formulas (3.12) and (3.13) should read \(A(X,Y)+A(Y,X)=0\) and \(C(X,Y)+C(Y,X)=\cdots\) respectively.