Differential geometry of tensor product immersions. II (Q1892332)

Let \(E^m\) be the \(m\)-dimensional Euclidean space with the canonical Euclidean inner product. The author defines the notion of tensor product map \(f \otimes h : M \to E^m \times E^{m'} \equiv E^{mm'}\) associated with any two maps \(f : M \to E^m\) and \(h : M \to E^{m'}\) of a given Riemannian manifold \((M,g)\). In the first part of this series [ibid. 11, No. 4, 345-359 (1993; Zbl 0824.53051)] the author proves that the tensor product immersion \(f_1 \otimes \cdots \otimes f_{2k}\) of \(2k\) isometric spherical immersions of a Riemannian manifold \(M\) in Euclidean space is of \(\ell\)-type with \(\ell \geq k\) and classifies tensor product immersions \(f_1 \otimes \cdots \otimes f_{2k}\) which are of \(k\)-type [for the notion of isometric immersion of \(k\)-type, see the author's book [Total mean curvature and submanifolds of finite type, World Scientific (1984; Zbl 0537.53049)]. In this paper, the author proves the following theorem: Let \(f : S^n(r) \to E^{n + 1}\) be an ordinary imbedding of an \(n\)-sphere of radius \(r\) into \(E^{n + 1}\) and \(h : S^n(r) \to S^{n (n +3)/2 - 1} (a) < E^{n (n + 3)/2}\) be a Veronese immersion. Then, for any open subset \(M\) of \(S^n(r)\), the tensor product immersion \(f_1 \otimes \cdots \otimes f_{2k}\), restricted to \(M\), is mass-symmetric and of \((k + 1)\)- type, where \(f_1 = \dots = f_{2k - 1} = f\) and \(f_{2k} = h\). Then, the author investigates tensor product immersions \(f_1 \otimes \cdots \otimes f_{2k}\) which are mass-symmetric and of \((k + 1)\)-type, and he proves four interesting theorems.