Submanifolds whose quadric representations satisfy \(\Delta \tilde x=B \tilde x+C\) (Q1373454)

Let \(x:M^n\rightarrow E^m\) be an \(n\)-dimensional submanifold immersed into Euclidean \(m\)-space. The quadric representation of the immersion \(x\) is defined as the smooth map \(\tilde{x}=xx^t:M^n\rightarrow SM(m)\), where \(x^t\) denotes the transpose of \(x\) and \(SM(m)\) is the space of \((m\times m)\)-real symmetric matrices. In [Proc. Am. Math. Soc. 114, 201-210 (1992; Zbl 0752.53030)], \textit{I. Dimitrić} made a deep study of this map and established some general results about the quadric representation, in particular those relative to the condition of \(\tilde{x}\) being of finite type. In the paper under review, the author studies submanifolds in \(E^m\) whose quadric representations satisfy the condition \(\Delta\tilde{x}=B\tilde{x}+C\), where \(\Delta\) stands for the Laplacian on \(M^n\), and \(B\) and \(C\) are two constant matrices. In particular, in the case where \(M^n\) has parallel mean curvature it is shown that it must be (an open piece of) an \(n\)-dimensional Euclidean space or be contained in a quadric hypersurface. It is also shown that the only \(n\)-dimensional submanifolds in \(E^m\) satisfying \(\Delta\tilde{x}=C\) are \(n\)-dimensional Euclidean spaces. Related results for the case of semi-Riemannian surfaces in the three-dimensional Lorentzian space forms can be found in [\textit{A. Ferrández, P. Lucas} and \textit{M. A. Meroño}, Houston J. Math. 22, 533-546 (1996)].