Normal space (to a surface)

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at a point $P$

The orthogonal complement $N_PF$ to the tangent space $T_PF$ (see Tangent plane) of the surface $F^m$ in $V^n$ at $P$. The dimension of the normal space is $n-m$ (the codimension of $F$). Every one-dimensional subspace of it is called a normal to $F$ at $P$. If $F$ is a smooth hypersurface, then it has a unique normal at every of its points.



[a1] W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)
[a2] B.-Y. Chen, "Geometry of submanifolds" , M. Dekker (1973)
How to Cite This Entry:
Normal space (to a surface). Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article