# Curvature lines, net of

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 |

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Curvature lines, net of.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Curvature_lines,_net_of&oldid=43550