View source for 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|Curvature line]]). A net of curvature lines on $V_{n-1}$ is a [[Conjugate net|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.P. Eisenhart,   "Riemannian geometry" , Princeton Univ. Press  (1949)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Shulikovskii,   "Classical differential geometry in a tensor setting" , Moscow  (1963)  (In Russian)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.C. Hsiung,   "A first course in differential geometry" , Wiley  (1981)  pp. Chapt. 3, Sect. 4</TD></TR></table>