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Mapping, principal net of a

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in a domain of a mapping

An orthogonal net in a domain of an -dimensional manifold (which can be, in particular, a Euclidean space) that is mapped onto a net, also orthogonal, by a diffeomorphism of onto a domain in the same or another Riemannian manifold . The directions tangential to the lines of the principal net of the mapping at the point are the principal directions of the ellipsoid of deformation of the induced mapping of the tangent space onto the tangent space . When , generally speaking, the principal net of a mapping is not holonomic. If the mapping is conformal, then any orthogonal net in the domain serves as a principal net.

References

[1] V.V. Ryzhkov, "Differential geometric point correspondences between spaces" Itogi Nauk. Ser. Mat. Geom. 1963 , 3 (1965) pp. 65–107 (In Russian)
How to Cite This Entry:
Mapping, principal net of a. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mapping,_principal_net_of_a&oldid=17526
This article was adapted from an original article by V.T. Bazylev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article