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Minimal Surfaces in Three-Dimensional Riemannian Manifold Associated with a Second-Order ODE

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Abstract

We show that a surface corresponding to a first-order ODE is minimal in three-dimensional Riemannian manifold which is determined by the first prolongation of \({\text {d}}y/\mathrm{d}x=p(x,y)\) if and only if \(p_{yy}=0\). Accordingly, any linear first-order ODE describes a minimal surface which is not necessarily totally geodesic.

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Authors and Affiliations

  1. Department of Mathematics, Akdeniz University, 07058, Antalya, Turkey

    T. Bayrakdar & A. A. Ergin

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  1. T. Bayrakdar
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  2. A. A. Ergin
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Correspondence to T. Bayrakdar.

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Bayrakdar, T., Ergin, A.A. Minimal Surfaces in Three-Dimensional Riemannian Manifold Associated with a Second-Order ODE. Mediterr. J. Math. 15, 183 (2018). https://doi.org/10.1007/s00009-018-1229-2

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  • DOI: https://doi.org/10.1007/s00009-018-1229-2

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