Generic Intersection of non-planar Surfaces in R^4

Click For Summary
SUMMARY

The discussion centers on the intersection of two planar surfaces in R^4, emphasizing that they generically intersect at isolated points rather than along lines. The concept of transversality is crucial, as it dictates that if the intersection is transverse, the resulting intersection forms a submanifold with a dimension calculated by the formula (dim A + dim B - dim M). In this case, the dimension is zero, confirming that intersections occur at discrete points. The conversation also touches on the Poincaré dual of the intersection form in cohomology, highlighting its integer-valued nature.

PREREQUISITES
  • Understanding of R^4 geometry
  • Familiarity with transversality in differential geometry
  • Knowledge of cohomology and Poincaré duality
  • Basic concepts of manifolds and submanifolds
NEXT STEPS
  • Study the concept of transversality in differential topology
  • Explore Poincaré duality in algebraic topology
  • Investigate the properties of submanifolds in R^n
  • Learn about intersection theory in higher-dimensional spaces
USEFUL FOR

Mathematicians, particularly those specializing in topology and geometry, as well as students and researchers interested in the intersection theory of manifolds in higher dimensions.

Bacle
Messages
656
Reaction score
1
Hi, everyone:

How do we show that 2 planar surfaces in R^4 intersect at points (possibly empty

sets of points, but not in lines, etc.).

I am curious to see how we justify the Poincare dual of the intersection form in

cohomology being modular, i.e., integer-valued.?

I am confused because the same does not seem to apply to, e.g., lines, which,

when embedded in R^2 or R^3 and R^4 , intersect (if at all) at points.

Thanks.
 
Physics news on Phys.org
It's transversality. If the intersection is transverse, they intersect in a submanifold with the appropriate dimension (dim A +dim B - dim M). Here the dimension is 0, so they generically intersect in isolated points, if at all.
 
Just to say thanks, Zhentil; you have been very helpful.
 

Similar threads

Undergrad Fixing an orientation for a connected smooth surface
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Eccentric anomaly of ellipse-circle intersections
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Get geodesics & parallel transport on 2D surfaces (discrete timestep)
  • · Replies 12 ·
Replies
12
Views
2K
Definition of Normal (Intersection) Without Using a Metric
  • · Replies 11 ·
Replies
11
Views
4K
Undergrad Follow-up Einstein Definition of Simultaneity for Langevin Observers
  • · Replies 4 ·
Replies
4
Views
1K
Intersection Form in 4-D. Followup.
  • · Replies 20 ·
Replies
20
Views
6K
Graduate Vanishing of an integral of a divergence over a closed surface
  • · Replies 1 ·
Replies
1
Views
1K
Topologising RP2 using open sets in R3
  • · Replies 5 ·
Replies
5
Views
4K
Undergrad On the Gaussian Curvature of time-like surfaces
  • · Replies 2 ·
Replies
2
Views
3K
Line of intersection of two planes
  • · Replies 9 ·
Replies
9
Views
3K