Definition of Normal (Intersection) Without Using a Metric

[Bacle]

Hi, Everyone:

I am trying to understand the meaning of a statement that two embedded manifolds
intersect normally*. The statement is made in a context in which any choice or existence
of a metric is not made explicit, nor--from what I can tell-- implicitly either.

If there was a choice of metric m, it seems reasonable to conclude that M,N
intersect normally if the tangent spaces at points of intersection are normal to
each other, i.e., if p is in the intersection, then TpM is orthogonal to TpN , with
respect to the metric given by m. Without a choice of m, it seems difficult to
see how to define normality of intersection. .

Is there then,a definition of normal intersection that does not make reference, or
does not make use of, the existence of a metric?

Thanks.

* More precisely, the statement is that M intersects N normally along DelM, the (manifold)
boundary of M

 

[quasar987]

Could it be that they just mean "transversally"?

 

[Bacle]

Thanks, Quasar, but i don't think so, since the author makes mention in another paragraph of manifolds being transverse; I would imagine s/he would have used
transverse in the paragraph I quoted.

 

[quasar987]

What is the manifold in which they are embedded?

 

[Bacle]

Quasar:
They are both embedded in S^4, the 4-sphere.

 

[quasar987]

Well there is a canonical metric on S^4! Namely, the metric you get when you embed S^4 in the usual way and then restrict the euclidean metric to it. So when no metric is specified, it is safe to assume they mean the canonical metric.

 

[lavinia]

Hi, Everyone:

I am trying to understand the meaning of a statement that two embedded manifolds
intersect normally*. The statement is made in a context in which any choice or existence
of a metric is not made explicit, nor--from what I can tell-- implicitly either.

If there was a choice of metric m, it seems reasonable to conclude that M,N
intersect normally if the tangent spaces at points of intersection are normal to
each other, i.e., if p is in the intersection, then TpM is orthogonal to TpN , with
respect to the metric given by m. Without a choice of m, it seems difficult to
see how to define normality of intersection. .

Is there then,a definition of normal intersection that does not make reference, or
does not make use of, the existence of a metric?

Thanks.

* More precisely, the statement is that M intersects N normally along DelM, the (manifold)
boundary of M

Normal intersection of two submanifolds means that at every point of the intersection the tangent spaces of the two submanifolds span the tangent space of the ambient manifold.

 

[quasar987]

Normal intersection of two submanifolds means that at every point of the intersection the tangent spaces of the two submanifolds span the tangent space of the ambient manifold.

That's the definition of transverse intersection, no?

 

[lavinia]

That's the definition of transverse intersection, no?

yes but I found a paper by Smale where he defines normal intersection in this way.

http://archive.numdam.org/ARCHIVE/A...63_3_17_1-2_97_0/ASNSP_1963_3_17_1-2_97_0.pdf

 

[Bacle]

Quasar Wrote, in Part:

"Well there is a canonical metric on S^4! Namely, the metric you get when you embed S^4 in the usual way and then restrict the euclidean metric to it. So when no metric is specified, it is safe to assume they mean the canonical metric."

True, but the layout/format of the article seemed to be purely topological , i.e., did not
make use of differential-topology-type techniques.

 

[quasar987]

Then, possibly the authors just mean "transverse", as lavinia suggested.

 

[Bacle]

Thanks to both; sorry for the dead-end chase.