Tutorial 2 and 3—Global Analysis
28.09/ 5.10. 2021
1. For i = 1, ...n let (Mi, Ai) be a smooth manifolds. Suppose M := M1 × ... × Mn
is endowed with the product topology. Then show that
A := {(U1 × ... × Un, u1 × ... × un) : (Ui, ui) ∈ Ai}
defines a smooth atlas on M and that the projections pri : M → Mi are smooth.
Moreover show that, if N is a smooth manifold and fi : N → Mi smooth functions,
then there exists a unique smooth function f : N → M such that pri ◦ f = fi and
that this property characterizes the smooth manifold structure on M uniquely.
2. Suppose (Mi, Ai) are smooth manifolds for i ∈ I, where I is countable. Consider
the disjoint union
M := i∈IMi = ∪i∈I{(x, i) : x ∈ Mi}
endowed with the disjoint union topology and denote by inji : Mi → M the canonical
injections (inji(x) = (x, i)). Show that A := ∪i∈IAi defines a smooth atlas
on M and that the injections inji are smooth. Moreover, show that for any smooth
manifold N and smooth functions fi : Mi → N, there exists a unique smooth function
f : M → N such that f ◦ inji = fi and show that this property characterizes
the smooth manifold structure on M uniquely.
3. Suppose U ⊂ Rm
is open and f : U → Rn
a smooth map such that Dxf : Rm
→
Rn
is of rank r for all x ∈ U.
Show that for any x0 ∈ U there exists a diffeomorphism φ between an open neighbourhood
of x0 and an open neighbourhood of 0 ∈ Rm
and a diffeomorphism ψ
between an open neighbourhood of y0 = f(x0) and an open neighbourhood of 0 in
Rn
such that the locally defined map
ψ ◦ f ◦ φ−1
: Rr
× Rm−r
→ Rr
× Rn−r
has the form (x1, ..., xr, ..., xm) → (x1, ..., xr, 0, ..., 0).
Hint: The idea is that f locally around x0 looks like Dx0 f, which is a linear map
Rm
→ Rn
of rank r, which up to a basis change has the form (x1, .., xm) →
(x1, ..., xr, 0, ..., 0).
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(a) Set E2 := ker(Dx0 f) ⊂ Rm
and E1 := E⊥
2 , and F1 := Im(Dx0 f) ⊂ Rn
and
F2 := F⊥
1 . Decompose
Rm
= E1 ⊕ E2 and Rn
= F1 ⊕ F2,
and consider f as a map f = (f1, f2) : E1 ⊕ E2 → F1 ⊕ F2 defined on
U ⊂ E1 ⊕ E2 = Rm
.
(b) Show that φ : E1 ⊕ E2 → F1 ⊕ E2 given by
φ(x1
, x2
) = (f1(x1
, x2
) − f1(x1
0, x2
0), x2
− x2
0)
is a local diffeomorphism around x0 = (x1
0, x2
0) whose local inverse will be
the required map.
(c) Show that g := f ◦ φ−1
: F1 ⊕ E2 → F1 ⊕ F2 has the form
g(y1
, y2
) = (g1((y1
, y2
), g2((y1
, y2
)) = (y1
+ y1
0, g2(y1
, 0)).
Now ψ is easily seen to be...?
4. Suppose M and N are are manifolds of dimension m respectively n and let f :
M → N be a smooth map of constant rank r. Deduce from (1) that for any fixed
y ∈ f(M) the preimage f−1
(y) ⊂ M is a submanifold of dimension m − r in M.
5. Consider the Grassmannian of r-planes in Rn
:
Gr(r, n) := {E ⊂ Rn
: E is a r-dimensional subspace of Rn
}.
Denote by Str(Rn
) the set of r-tuples of linearly independent vectors in Rn
. Identifying
an element X ∈ Str(Rn
) with a n × r matrix
X = (x1
, ...., xr
) xi
∈ Rn
,
shows that Str(Rn
) equals the subset of rank r matrices in the vector space Mn×r(R),
which we know from Tutorial 1 is an open subset. Write
π : Str(Rn
) → Gr(r, n)
for the natural projection given by π(X) = span(x1
, ..., xr
) and equip Gr(r, n) with
the quotient topology with respect to π.
(a) Fix E ∈ Gr(r, n) and let F ⊂ Rn
be a subspace of dimension n − r such that
Rn
= E ⊕ F. Show that
U(E,F) = {W ∈ Gr(r, n) : W ∩ F = {0}} ⊂ Gr(r, n)
is an open neighbourhood of E.
(b) Show that any element W ∈ U(E,F) determines a unique linear map
W : E → F
such that its graph equals W, i.e. W = {(x, Wx) : x ∈ E}.
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(c) Show that the map uE,F : U(E,F) → Hom(E, F) given by uE,F (W) = W is a
homeomorphism.
(d) Show that
A := {(U(E,F), u(E,F)) : E, F ⊂ Rn
complimentary subspaces of dimension r resp. n−r}
is a smooth atlas for Gr(r, n).
6. For a topological space M denote by C0
(M) the vector space of continuous realvalued
functions f : M → R. Any continuous map F : M → N between
topological spaces M and N induces a map F∗
: C0
(N) → C0
(M) given by
F∗
(f) := f ◦ F : M → R.
(a) Show that F∗
is linear.
(b) If M and N are (smooth) manifolds, show that F : M → N is smooth ⇐⇒
F∗
(C∞
(N)) ⊂ C∞
(M).
(c) If F is a homeomorphism between (smooth) manifolds, show that F is a diffeomorphism
⇐⇒ F∗
is an isomorphism.