Math 545 Topology and Geometry of Manifolds Winter 2000
Assignment #3
Due 1/26/2000
I. Required problems.
1. Let Vbe a finite-dimensional vector space with its standard smooth structure.
Show that for every a∈Vthere is a natural (basis-independent) isomorphism V
−→ TaVsuch for any linear map L:V−→ Wthe following diagram commutes:
WT
L(a)W
-
∼
=
VT
aV
-
∼
=
?
L
?
L∗
[Hint: look at the definition of the derivation e
Vafor a geometric tangent vector
Va∈Rn
a.]
2. Let Mnn(R)denotethen2-dimensional vector space of n×nreal matrices. For
any A∈Mnn(R), define
eA=
∞
X
k=0
1
k!Ak.
(a) Show that the series converges uniformly on any bounded subset of Mnn(R),
and defines a smooth map exp: Mnn(R)−→ Mnn (R) defined by exp(A)=eA.
(b) Using the push-forward formula for tangent vectors to curves, compute
exp∗:TAMnn(R)−→ Texp AMnn(R). (Use the result of Problem 1 to iden-
tify both tangent spaces with Mnn (R)itself.)
3. Let M1,... ,M
kbe smooth manifolds. For any choices of points pi∈Mi,i=
1,... ,k, show that the projection maps πj:M1× ··· × Mk−→ Mjyield an
isomorphism
π1⊕···⊕ πk:T(p1,... ,pk)(M1×··· ×Mk)−→ Tp1M1⊕··· ⊕TpkMk.
[Using this isomorphism, we will routinely identify TpMas a subspace of T(p,q)(M×
N).]
4. Suppose F:M−→ Nis a smooth map. Show that F∗:TM −→ TN is smooth.