- #1

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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.

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- Thread starter WWGD
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In summary, 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.

- #1

- 7,314

- 11,126

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.

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- #2

quasar987

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- #3

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Thanks, Quasar; any chance you have a ref?

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quasar987

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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

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

Last edited:

- #5

Bacle

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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

x-axis.

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 x-axis as an example of a non-transverse 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 .

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

x-axis.

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 x-axis as an example of a non-transverse 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 .

Last edited by a moderator:

A transversal intersection of manifolds is when two or more manifolds intersect at a common point, and their tangent spaces at that point do not have any common vectors. This means that the manifolds intersect at a right angle, or "transversally".

No, the transversal intersection of manifolds is not always a manifold. It depends on the dimension and properties of the manifolds involved. In some cases, the intersection may not be smooth enough to be considered a manifold.

One example is a sphere intersecting with a plane. Another example is a torus intersecting with a line. In both cases, the intersection is transversal and results in a new manifold.

The transversal intersection of manifolds is useful in various fields of mathematics, such as differential geometry and topology. It allows for the study of the local properties of manifolds and their intersections, which can provide insight into the global properties of the manifolds.

Yes, the transversal intersection of manifolds has applications in physics, specifically in the field of mechanics. It is used to study the motion of particles and systems in a way that is independent of the coordinate system, making it a useful tool for understanding complex physical systems.

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