SOLUTIONS TO EXERCISES
CHAPTER I
A. Manifolds
A.1. First take a covering cfw_V,,l of A by open relatively compact sets V disjoint from B. Then take a covering cfw_VsB,J f the closed set o M U., V. by open relatively compact sets VB disjoint from A. The c
CHAPTER
LIE GROUPS AND
II
LIE ALGEBRAS
A Lie group is, roughly speaking, an analytic manifold with a group structure
such that the group operations are analytic. Lie groups arise in a natural way as
transformation groups of geometric objects. For example,
CH,.)
EXERCISES AND FURTHER RESULTS
A. On the Geometry of Lie Groups
g

1. Let G be a Lie group, L(x) and R(x), respectively the left translation
xg, and the right translation g  gx. Prove:
(i) Ad(x) = dR(x)
o dL(x)e = dL(x),
o dR(x 1)e.
(ii) If J is
CH,I)
EXERCISES AND FURTHER RESULTS
A. Manifolds
1. Let M be a paracompact manifold, A and B disjoint closed subsets
of M. Then there exists a function f E C(M) such that f  I on A,
f0 on B.
2. Let M be a connected manifold and p, q two points in M. The