Extension of minimal immersions of spheres into spheres
The purpose of the present study is to get isometric minimal immersions of \(S^{m+k}(1)\) into spheres which are extensions of isometric minimal immersions of \(S^ m(1)\) into spheres. The idea of extension is as follows. Let \(P\) be a projection \(R^{m+k+1}\to R^{m+1}\) where \(R^{m+1}\) is considered as a subspace of \(R^{m+k+1}\), \(T\) be a symmetric tensor of degree \(s\) on \(R^{m+1}\) and \(\tilde v_i\) be vectors in \(R^{m+k+1}\). Then the symmetric tensor \(\tilde T\) on \(R^{m+k+1}\) defined by \(\tilde T(\tilde v_1,\ldots,\tilde v_s)=T(P\tilde v_1,\ldots,P\tilde v_s)\) is called the extension of \(T\). If isometric minimal immersions \(S^m(1)\to S^{n-1}(r)\) are induced by harmonic polynomials (spherical harmonics) of degree \(s\) on \(S^m(1)\), they are denoted by \(f_{m,s}\), and the vector space \(W_2\) and the compact convex body \(L\) of [\textit{M. P. do Carmo} and \textit{N. R. Wallach}, Ann. Math. (2) 93, 43--62 (1971; Zbl 0218.53069)] are denoted by \(W(m,s)\) and \(L(m,s)\). Let us take an immersion \(f_{m,s}\). Then guided by the idea of extension of tensors an extension of \(f_{m,s}\) is defined and denoted by \(\mathoperator{Ext}_kf_{m,s}\). As a result we get an injective homomorphism \(\Lambda: W(m,s)\to W(m+k,s)\) such that \(\Lambda L(m,s)=L(m+k,s)\cap \Lambda (m,s)\). The effect of the extension on geometric properties of isometric minimal immersions is studied.
scientific article; zbMATH DE number 4147041