Tutorial 3-4—Global Analysis
1. We have seen in the first tutorial that Homr(Rn
, Rm
) is a submanifold of Hom(Rn
, Rm
)
of dimension r(n + m − r) in. For X ∈ Homr(Rn
, Rm
) compute the tangent space
TXHomr(Rn
, Rm
) ⊂ TXHom(Rn
, Rm
) ∼= Hom(Rn
, Rm
).
2. We have seen in the first tutorial that the Grassmannian manifold Gr(r, n) can be
realized as a submanifold of Hom(Rn
, Rn
) of dimension r(n−r). For E ∈ Gr(r, n)
compute the tangent space
TEGr(r, n) ⊂ TEHom(Rn
, Rn
) ∼= Hom(Rn
, Rn
).
3. Consider the general linear group GL(n, R) and the special linear group SL(n, R).
We have seen that they are submanifolds of Mn(R) = Rn2
(even so called Lie
groups) and that TIdGL(n, R) ∼= Mn(R) = Rn2
.
(a) Compute the tangent space TIdSL(n, R) of SL(n, R) at the identity Id.
(b) Fix A ∈ SL(n, R) and consider the conjugation conjA : SL(n, R) → SL(n, R)
by A given by conjA(B) = ABA−1
. Show that conjA is smooth and compute
the derivative TIdconjA : TIdSL(n, R) → TIdSL(n, R).
(c) Consider the map Ad : SL(n, R) → Hom(TIdSL(n, R), TIdSL(n, R)) given
by Ad(A) := TIdconjA. Show that Ad is smooth and compute TIdAd.
4. Consider Rn
equipped with the standard inner product of signature (p, q) (where
p + q = n) given by
x, y :=
p
i=1
xiyi −
n
i=p+1
xiyi
and the group of linear orthogonal transformation of (Rn
, ·, · ) given by
O(p, q) := {A ∈ GL(n, R) : Ax, Ay = x, y ∀x, y ∈ Rn
}.
(a) Show that
O(p, q) = {A ∈ GL(n, R) : A−1
= Ip,qAt
Ip,q},
where Ip,q =
Idp 0
0 −Idq
, and that O(p, q) is a submanifold of Mn(R). What
is its dimension?
1
2
(b) Show that O(p, q) is a subgroup of GL(n, R) with respect to matrix multiplication
µ and that µ : O(p, q) × O(p, q) → O(p, q) is smooth (i.e. that O(p, q)
is a Lie group.)
(c) Compute the tangent space TIdO(p, q) of O(p, q) at the identity Id.