Intrinsic obstructions to the existence of isometric minimal immersions. (Q1858306)

After surveying the results about Chern's problem to find conditions for a given Riemannian manifold to be realizable as a minimal submanifold of an Euclidean space (see \textit{J. L. Barbosa} and \textit{M. P. do Carmo} [An. Acad. Brasil. Ci. 50, 451--454 (1978; Zbl 0402.53004); \textit{S.-S. Chern} and \textit{R. Osserman}, Geometry, Proc. Symp., Utrecht 1980, Lect. Notes Math. 894, 49--90 (1981; Zbl 0477.53056), \textit{M. P. do Carmo} and \textit{M. Dajczer}, Bol. Soc. Bras. Mat. 12, No. 1, 113--121 (1981; Zbl 0604.53023), \textit{B.-Y. Chen}, Arch. Math. 60, No. 6, 568--578 (1993; Zbl 0811.53060)], the author provides some new necessary conditions. Let \(M^n\), \(n\geq 3\), be an \(n\)-dimensional Riemannian manifold with negative Ricci curvature and consider the Riemannian metric \(\langle,\rangle_*=-\text{Ric}\). It is proved that a necessary condition for \(M^n\) to admit an isometric minimal immersion (resp. with flat normal bundle) into an Euclidean space is the following \[ \inf K_*< \frac{3n\tau^2-2\|\text{Ric} \|^2-(n-1)\Delta\tau} {2\bigl( \tau^2-\|\text{Ric} \|^2\bigr)}, \quad\text{(resp. }\inf K_*\leq 1), \] where \(K_*\) is the sectional curvature of \(\langle,\rangle_*\), \(\tau\) is the scalar curvature of \(M^n\), \(\|\text{Ric} \|\) is the length of the Ricci tensor, and \(\Delta\) is the Laplacian operator of \(M^n\). For the case of \(n=2\) in addition to [\textit{E. Calabi}, Ann. Math. Stud. 30, 77--85 (1953: Zbl 0053.05103), \textit{H. B. Lawson jun.} Lectures on minimal submanifolds. Vol. I. Berkeley, California: Publish Perish (1980; Zbl 0434.53006)], given in the references, also a paper of the reviewer [Uch. Zap. Tartu. Gos. Univ. 129, Tr. Mat. Mekh. 3, 74--89; 90--102 (1962; Zbl 0156.41404)] can be indicated. In the latter it is proved that if \(g\) is the metric of a minimal surface in a four-dimensional Euclidean space whose indicatrices of normal curvature are circles, then \(\root 3\of{-\frac 12 Kg}\) has zero Gaussian curvature.