Meaning of Transversality For Different Dimensions
Suppose we have transverse maps and where are manifolds with dimensions respectively.
The meaning of transversality differs a lot depending on the relative dimensions of and . The relationship between transversality and tangency is clearest when .
We can consider three separate cases:
- When, it is impossible for the image of and 's tangent spaces to span 's tangent space at any point. Thus any intersection between and cannot be transverse. However, non-intersecting manifolds vacuously satisfy the condition, so can be said to intersect transversely.
- When, the image of and 's tangent spaces must sum directly to 's tangent space at any point of intersection. Their intersection thus consists of isolated signed points, i.e. a zero-dimensional manifold.
- When this sum needn't be direct. In fact it cannot be direct if and are immersions at their point of intersection, as happens in the case of embedded submanifolds. If the maps are immersions, the intersection of their images will be a manifold of dimension .
Read more about this topic: Transversality (mathematics)
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