Klyachin, V. A. New examples of tubular minimal surfaces of arbitrary codimension. (English. Russian original) Zbl 0987.53004 Math. Notes 62, No. 1, 129-131 (1997); translation from Mat. Zametki 62, No. 1, 154-156 (1997). From the text: Let \(F\) be a compact manifold, \((\alpha,\beta)\) an interval and let us consider a \(C^2\)-smooth function \(r(t)\) defined on \((\alpha,\beta)\). Then we define a \(p\)-dimensional surface \(M\) by the \(C^2\)-smooth embedding \(u: F\times (\alpha,\beta)\to \mathbb{R}^{n+1}\), \(u(y,t)= r(t)R(y)+ te_0\), where \(R: F\to S^{n-1}\subset\{0\}\times \mathbb{R}^n\subset \mathbb{R}^{n+1}\) is a \(C^2\)-smooth embedding of a \((p-1)\)-dimensional manifold \(F\).Our main result is the following theorem. A surface \(M\) of the above form is a minimal tubular submanifold with projection \((\alpha,\beta)\) if and only if \(R: F\to S^{n-1}\) is a minimal embedding in the sphere and the function \(r(t)\) satisfies the differential equation \(r''(t)r(t)= (p-1)(1+ r^{\prime 2}(t))\). MSC: 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces Keywords:minimal tubular submanifold; minimal submanifold sphere PDFBibTeX XMLCite \textit{V. A. Klyachin}, Math. Notes 62, No. 1, 129--131 (1997; Zbl 0987.53004); translation from Mat. Zametki 62, No. 1, 154--156 (1997) Full Text: DOI References: [1] A. D. Vedenyapin and V. M. Miklyukov,Mat. Sb. [Math. USSR-Sb.],131, No. 2, 240–250 (1986). [2] V. M. Miklyukov and V. G. Tkachev,Mat. Sb. [Math. USSR-Sb.],180, No. 9, 1278–1295 (1989). [3] V. A. Klyachin,Sibirsk. Mat. Zh. [Siberian Math. J.],33, No. 4, 201–205 (1992). [4] A. T. Fomenko,Variational Methods in Topology [in Russian], Nauka, Moscow (1982). · Zbl 0526.58012 [5] S. Kobayasi and K. Nomizu,Foundations of Differential Geometry, Vol. 2, Interscience Publ., New York-London (1981). [6] J. Simons,Ann. of Math.,88, No. 2, 62–105 (1968). · Zbl 0181.49702 · doi:10.2307/1970556 [7] H. Gauchman,Trans. Amer. Math. Soc.,298, No. 2, 779–791 (1986). · doi:10.1090/S0002-9947-1986-0860393-5 [8] S.-S. Chern, M. do Carmo, and S. Kobayasi, in:Functional Analysis and Related Fields, Springer-Verlag, Berlin-New York (1970), pp. 59–75. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.