50 XVI DIFFERENTIAL MANIFOLDS

8. Let X, Y be two differential manifolds, / : X-> Y a mapping of class C°°, and U a
connected open subset of X. If r is the least upper bound of rk*(/) as x runs over U,
show that r is finite and that the set of points x e U at which rk*(/) — r is open. Deduce
that the set of points at which /is a subimmersion is a dense open subset of X (argue
by contradiction). If /is an open mapping, the set of points at which /is a submersion
is dense in X.

9. Let X, Y be two differential manifolds, / : X -> Y a mapping of class C1 . If Z is a sub-
manifold of Y, the mapping/is said to be transversal over Zatx ef~i(Z) if the tangent
space T/(Jo(Y) is the sum of T/(3e)(Z) and T*(/)(TX(X)), and /is said to be transversal
over Z if this condition is satisfied for all x e/-J(Z). If so, then/'1® is a submanifold
of X, and for each x e/"1® the tangent space T^/'HZ)) is the inverse image under
T»(/) of T/(*)(Z), (Since the question is a local one, we may take Z to be a submanifold
given by an equation g(y) = 0, where g : Y ->RP is a submersion; consider the com-
posite mapping g of.)

In particular, if X and Z are submanifolds of Y, we say that X and Z are transversal
at a point x e X n Z if the canonical injection of X into Y is transversal over Z at x,
or equivalently if T,(Y) = TX(X) + TX(Z), which is symmetrical in X and Z, The sub-
manifolds X and Z are said to be transversal if they are transversal at all points
jc e X n Z; in that case, X n Z is a submanifold of Y.

10. Let / : X -» Z and g : Y -*• Z be two mappings of class C00 , and consider their product
/x g : X x Y->Z x Z, which is also of class C°°. Show that/ x g is transversal over
the diagonal A of Z x Z if and only if, for each pair (x, y) e X x Y such that/(x) = g( y),
we have

W Tz(Z) = Tx(/)(Tx(X)) + Ty(^)(Ty(Y)),

where z =/(x) = g(y). This condition is always satisfied if either /or g is a submersion.
When condition (*) is satisfied, the set of points (x, y) e X x Y such that/(*) = g(y)
is a submanifold of X x Y, which is called the fiber product ofX and Y over Z and is
written X x z Y. The tangent space at the point (x, y) e X x z Y to the fiber product is
the subspace of T*(X) x Ty(Y) consisting of the pairs (h, k) such that

In this situation, /and g are said to be transversal mappings into Z. Show that if
/is a submersion (resp. an immersion, resp. a subimmersion), then so is the restriction
Xx2Y->Yofpr2.

11. Let Y be a differential (resp. real-analytic, resp. complex-analytic) manifold, X a
Hausdorff topological space, and p : X-> Y a mapping with the following property:
For each x e X there exists an open neighborhood V of x such that p \ V is a homeo-
morphism of the subspace V onto a submanifold of Y.

(a) Show that X is locally connected, that each point of x has a closed neighborhood
which is homeomorphic to a closed ball in R", and that for each y e Y the fiber p~l(y)
is a discrete subspace of X.

(b) Let 95 be a denumerable basis for the topology of Y. A pair (W, U) is said to be
distinguished if U eS5 and if W is a connected component of /rHXJ) such that W is
compact and metrizable and p \ W is a homeomorphism of W onto a subspace of Y. n\ — m, and the jets of the monomials xt-^x* *e,