Mapping, principal net of a

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in a domain of a mapping

An orthogonal net in a domain $ G $ of an $ n $- dimensional manifold $ M $( which can be, in particular, a Euclidean space) that is mapped onto a net, also orthogonal, by a diffeomorphism $ f: G \rightarrow G ^ \prime $ of $ G $ onto a domain $ G ^ \prime $ in the same or another Riemannian manifold $ M ^ \prime $. The directions tangential to the lines of the principal net of the mapping at the point $ x \in G $ are the principal directions of the ellipsoid of deformation of the induced mapping $ f _ \star : T _ {x} \rightarrow T _ {f(} x) ^ \prime $ of the tangent space $ T _ {x} $ onto the tangent space $ T _ {f(} x) ^ \prime $. When $ n > 2 $, generally speaking, the principal net of a mapping is not holonomic. If the mapping $ f $ is conformal, then any orthogonal net in the domain $ G $ serves as a principal net.


[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:,_principal_net_of_a&oldid=47756
This article was adapted from an original article by V.T. Bazylev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article