Is the Transversal Intersection of Manifolds a Manifold?by WWGD Tags: intersection, manifold, manifolds, transversal 

#1
Jul211, 06:00 PM

P: 398

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. 



#2
Jul311, 08:36 PM

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PF Gold
P: 4,768

Yes, according to the "canonical form theorem" for a transverse intersection, if X,Y are submanifolds of the nmanifold 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.




#3
Jul411, 01:45 PM

P: 398

Thanks, Quasar; any chance you have a ref?




#4
Jul411, 09:18 PM

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HW Helper
PF Gold
P: 4,768

Is the Transversal Intersection of Manifolds a Manifold?
See the book Differential manifolds by Antoni Kosinski where the stronger result is proved that actually, there is a coordinate chart around p in which X corresponds to R^r x {0} while Y corresponds to {0} x R^s, so what X n Y, of course, corresponds to {0} x R^k x {0} (where r=dim(X), s=dim(Y), and k=(r+s)).
But surely the theorem can also be found in Differential Topology by Guillemin & Pollack and possibly in the book of the same name by M. Hirsh 



#5
Jul1411, 07:13 PM

P: 662

WWGD:
Not to quibble too much, but I have seen two main definitions of transversality used, and I wondered which one you are using: 1) First and weaker of the two, states that if submanifolds S,S' of ambient M intersect transversely at p, then the (vector space) sum of the tangent spaces TpS and TpS' equals TpM , i.e., the sum spans the tangent space of the ambient manifold. 2) The second and stronger one (stronger in that it excludes some cases of 1) , is that each point p of intersection has a neighborhood Up with Phi(Up)= (x1,x2,..,xn,0,0,..,0) (Phi is a chart map). This excludes, e.g., Sin(1/x) and the xaxis. BTW, the Mathworld entry "http://mathworld.wolfram.com/HomologyIntersection.html" seems to agree in a weaker sense, in that it states that cycles (homology classes) that intersect transversely are also cycles ,and it gives the example of y=x^2 and the xaxis as an example of a nontransverse intersection in the sense 2) above, showing how the intersection is unstable, in that a small perturbation e.g., moving y=x^2 upwards changes the (algebraic) intersection number . 


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