# Is the Transversal Intersection of Manifolds a Manifold?

by WWGD
Tags: intersection, manifold, manifolds, transversal
 P: 311 Hi, All: Given manifolds M,N (both embedded in $R^n$, intersecting each other transversally, so that their intersection has dimension >=1 ( i.e. n -(Dim(M)-Dim(N)>1) is the intersection a manifold? Thanks.
 PF Patron HW Helper Sci Advisor P: 4,755 Yes, according to the "canonical form theorem" for a transverse intersection, if X,Y are submanifolds of the n-manifold M that intersect transversally "of dimension k", and if p is a point of intersection, there is a coordinate nbhd of p in M such that X n Y corresponds to R^k in R^n under the coordinate map.
 P: 311 Thanks, Quasar; any chance you have a ref?
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