\(D\)-lines on the surfaces of parallel mean curvature in arbitrary dimensional manifolds of constant curvature (Q2708399)
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scientific article
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | \(D\)-lines on the surfaces of parallel mean curvature in arbitrary dimensional manifolds of constant curvature |
scientific article |
Statements
17 April 2001
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\(D\)-lines
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Darboux lines
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parallel mean curvature
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constant mean curvature
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0.8977399
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0.8888932
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\(D\)-lines on the surfaces of parallel mean curvature in arbitrary dimensional manifolds of constant curvature (English)
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Darboux was the first to study the problem of determining the lines of a surface whose osculating sphere is tangent to the surface at each point. Such lines are called \(D\)-lines. In this paper the notion \(D\)-line on 2-dimensional surfaces of arbitrary codimension in spaces of constant curvature is defined. For totally umbilic and pseudo-umbilical surfaces the differential equation of \(D\)-lines is expressed in terms of the partial derivatives of the conformal parameter on \(M^2\) in the mean curvature normal direction. Certain classical results for \(D\)-lines on surfaces in \(E^3\) are generalized and some new ones obtained.
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