Transversality

In Linear algebra and differential Geometry, the property of transversality is a qualifier for the Intersection of subspaces or subvarieties. It is to some extent the opposite of the concept of tangency.

Two vectorial subspaces F , G of a vector Space E are known as transverses when $F+G=E$ . This condition can be rewritten, if necessary, in terms of Codimension:

{\ mathrm codim} F + {\ mathrm codim} G= {\ mathrm codim} (F \ course G)

Two subvarieties $M$ and $M'$ of a differential variety $P$ are transverse when, for any point $x$ of $M \ course N$ , tangent spaces are transverse:

T_xP=T_xM + T_xN

In the continuation, $m, N, p$ indicate respective dimensions of $M, NR, P$ .

Note:

the definition remains valid for the banachic varieties.
Two disjoined subvarieties are transverse.
If $m+n, then the condition of transversality cannot be only checked if the subvarieties M and P are disjoined.$

Theorem. If $m+n \ geq p$ , then a transverse intersection $M \ course N$ is a differential subvariety of dimension $m+n-p$ .

Example: two Surface S regular of space with three dimensions are transverse if and only if they do not have any tangential point. In this case, their intersection forms a regular Courbe (possibly vacuum).

Number of intersection

Generics

Theorem. If $M$ and $N$ are two subvarieties $C^k$ of respective size $m$ and $n$ , then there exists a $C^k$ diffeomorphism $H$ of $P$ , as near to the identity as wished in topology $C^k$ , such as $h (M)$ intersects $N$ transversely.

In general, two subvarieties are transversely intersected, even if it means to disturb one of them by a Isotopie.

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