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Curvature lines, net of

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An orthogonal net on a smooth hypersurface $V_{n-1}$ in an $n$-dimensional Euclidean space $E_n$ ($n\geq3$), formed by the curvature lines (cf. Curvature line). A net of curvature lines on $V_{n-1}$ is a conjugate net. E.g., if $V_2\subset E_3$ is a surface of revolution, the meridians and the parallels of latitude form a net of curvature lines. If $V_p\subset E_n$ ($2\leq p<n$) is a smooth $p$-dimensional surface with a field of one-dimensional normals such that the normal $[x,\mathbf n]$ of the field lies in the second-order differential neighbourhood of the point $x\in V_p$, then the normals of the field define curvature lines and a net of curvature lines on $V_p$, exactly as on $V_{n-1}$. However, a net of curvature lines on $V_p$ ($p<n-1$) need not be conjugate.

References

[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)


Comments

References

[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: http://www.encyclopediaofmath.org/index.php?title=Curvature_lines,_net_of&oldid=43550
This article was adapted from an original article by V.T. Bazylev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article