# Mapping, principal net of a

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.