Curvature lines, net of

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An orthogonal net on a smooth hypersurface in an -dimensional Euclidean space (), formed by the curvature lines (cf. Curvature line). A net of curvature lines on is a conjugate net. E.g., if is a surface of revolution, the meridians and the parallels of latitude form a net of curvature lines. If () is a smooth -dimensional surface with a field of one-dimensional normals such that the normal of the field lies in the second-order differential neighbourhood of the point , then the normals of the field define curvature lines and a net of curvature lines on , exactly as on . However, a net of curvature lines on () need not be conjugate.


[1] L.P. Eisenhart, "Riemannian geometry" , Princeton Univ. Press (1949)
[2] V.I. Shulikovskii, "Classical differential geometry in a tensor setting" , Moscow (1963) (In Russian)



[a1] C.C. Hsiung, "A first course in differential geometry" , Wiley (1981) pp. Chapt. 3, Sect. 4
How to Cite This Entry:
Curvature lines, net of. Encyclopedia of Mathematics. URL:,_net_of&oldid=16117
This article was adapted from an original article by V.T. Bazylev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article