Page 259 - Curvature and Homology

EXERCISES 241

is homeomorphiic with P,--the methods employed being essentially algebraic
geometric, that is, a knowledge of the classification of surfaces being necessary.)
D.l has been extended to all dimensions by R. L. Bishop and S. I. Goldberg
Pol*
E. Symmetric homogeneous spaces [26]
1. Let G be a Lie group and H a closed subgroup of G. The elements a, b E G
are said to be congruent modulo H if aH = bH. This is an equivalence relation
-the equivalence classes being left cosets modulo H. The quotient space G/H
by this equivalence relation is called a homogeneous space.
Denote by n : G-+ G/H the natural map of G onto G/H (n assigns to a E G
its coset modulo H). Since G and H are Lie groups G/H is a (real) analytic
manifold and * is an analytic map. H acts on G by right translations: (x,a) +- xn,
x E G, a E H. On the other hand, G acts on G/H canonically, since the left
translations by G of G commute with the action of H on G. The group G is a
Lie transformation group on G/H which is transitive and analytic, that is, for
any two points on G/H, there is an element of G sending one into the other.
Let o be a non-trivial involutary automorphism of G : 9 = I, a # I. Denote
by G, the subgroup consisting of all elements of G which are invariant under o
and let denote the component of the identity in Go. If His aclosed subgroup
of G with G: as its component of the identity, G/H is called a syrnmetrk homogeneous
Epace*
Let G/H be a symmetric homogeneous space of the compact and connected
Lie group G. Then, with respect to an invariant Riemannian metric on G/H
an invariant form (by G) is harmonic, and conversely.
In the first place, since G is connected it can be shown by averaging over G
that a differential form a on G/H invariant by G is closed. (Since G is transitive,
an invariant differential form is uniquely determined by its value at any point
of M). Let h be a Riemannian metric on G/H and denote by a*h the transform
ofhbya~G.Put

Then, g is a metric on G/H invariant by G. In terms of g, *a is also invariant
and therefore closed. Thus, a is a harmonic form on G/H.
2. Show that
P,, = U(n + 1)/ U(n) x U(1)
is a symmetric homogeneous space.
To see this, we define an involutory automorphism u of U(n + 1) by