View source for Mapping, principal net of a

''in a domain of a mapping''

An [[Orthogonal net|orthogonal net]] in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m0623101.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m0623102.png" />-dimensional [[Manifold|manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m0623103.png" /> (which can be, in particular, a Euclidean space) that is mapped onto a net, also orthogonal, by a [[Diffeomorphism|diffeomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m0623104.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m0623105.png" /> onto a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m0623106.png" /> in the same or another Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m0623107.png" />. The directions tangential to the lines of the principal net of the mapping at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m0623108.png" /> are the principal directions of the ellipsoid of deformation of the induced mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m0623109.png" /> of the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m06231010.png" /> onto the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m06231011.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m06231012.png" />, generally speaking, the principal net of a mapping is not holonomic. If the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m06231013.png" /> is conformal, then any orthogonal net in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062310/m06231014.png" /> serves as a principal net.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.V. Ryzhkov,   "Differential geometric point correspondences between spaces"  ''Itogi Nauk. Ser. Mat. Geom. 1963'' , '''3'''  (1965)  pp. 65–107  (In Russian)</TD></TR></table>