Two submanifolds 
and 
in an ambient space 
intersect transversally if,
for all 
,
 |
where the addition is in 
, and 
denotes the tangent map
of 
.
If two submanifolds do not intersect, then they
are automatically transversal. For example, two curves in 
are transversal only if they do not intersect
at all. When 
and 
meet transversally then 
is a smooth submanifold
of the expected dimension 
.
In some sense, two submanifolds "ought" to intersect transversally and, by Sard's theorem, any intersection can be perturbed to be transversal. Intersection in homology only makes sense because an intersection can be made to be transversal.
Transversality is a sufficient condition for an intersection to be stable after a perturbation. For example, the lines 
and 
intersect transversally,
as do the perturbed lines 
, and they intersect at
only one point. However, 
does not intersect 
transversally. It intersects in one point, while
intersects in either none or two points, depending
on whether 
is positive or negative.
When 
,
then a transversal intersection is an isolated point.
If the three spaces have an vector space orientation,
then the transversal condition means it is possible to assign a sign to the intersection.
If 
are an oriented basis for 
and 
are an oriented basis for 
, then the intersection is 
if 
is oriented in 
and 
otherwise.
More generally, two smooth maps 
and 
are transversal if whenever 
then 
.
