Curvature lines, net of
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.
|||L.P. Eisenhart, "Riemannian geometry" , Princeton Univ. Press (1949)|
|||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|
Curvature lines, net of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Curvature_lines,_net_of&oldid=16117