A property of conformal infinitesimal deformations of multidimensional surfaces in Riemannian space (Q1358473)

An infinitesimal deformation of a \(k\)-dimensional submanifold \(F^k\) of class \(C^3\) in an \(n\)-dimensional Riemannian space \(R^n\) is called conformal if the variation of the angle between any two directions at any point \(x\in F^k\) is zero and areally recurrent if \(\delta(d_x \sigma(G^2))=\varphi(x)d_x \sigma(G^2)\) holds for the variation of the area element \(d_x\sigma(G^2)\) of any two-dimensional surface drawn through any point \(x\), where \(\varphi\) is a certain function of a class \(C^2\) on \(F^k\) defined by the infinitesimal deformation. It is proved that conformal infinitesimal deformations of \(F^k\) in \(R^n\), \(k>2\), and they only, are areally recurrent infinitesimal deformations. All areally recurrent deformations of the hypersphere \(S^{n-1}\) in Euclidean space \(E^n\), \(n>3\), are described: each such deformation field is the sum of an infinitesimal isometry and a normal infinitesimal deformation of \(S^{n-1}\) in \(E^n\) (the latter in the sense of [\textit{B.-Y. Chen} and \textit{K. Yano}, J. Differ. Geom. 13, 1-10 (1978; Zbl 0403.53012)]).