J. Korean Math. Soc. 31
(1994),No.
4,pp.
629-632SCALAR CURVATURES OF INVARIANT METRICS*
JOON-SIK PARK AND WON-TAE
OH
1. Introduction
Let (M, g) be a Riemannian manifold. The relation between a (pointwise) conformal metric of the metric 9 and the scalar curvature of this new metric is investigated by Kazdan, Warner and Schoen (cf.
[1, 4]).
In this note, we consider the scalar curvatures of Riemannian metrics of three dimensional sphere 53. Since 53 and 5U(2) are diffeomorphic, we deal with 5U (2) in place of 53.
Throughout this paper, we investigate the relation between har- monic inner automorphisms of (5U(2), g) and the scalar curvature for a given left invariant Riemannian metric (cf. Theorem 1). Moreover, we completely classify the scalar curvature of (5U(2), g) for every left- invariant Riemannian metric 9 into positive, negative, zero scalar cur- vatures (cf. Theorem 2).
2. Scalar curvatures of left invariant metrics on 5U(2) Let M denote the Lie group 5U(2). We denote by g the Lie algebra of all left invariant vector fields on SU(2). The Killing form B of g satisfies B(X, Y) = 4 Trace(XY), (X, Y E g). We define an inner product ( , )0 on g by
( , )0
:=-B(X, Y), (X, Y E g). (1)
( , ) 0
determines a left invariant metric go on M. The following Lemma is known (c£. [5, Lemma 1.1, p. 154]):
Received July
24, 1993.* This research was supported by TGRC-KOSEF.
630 Joon-Sik Park and Won-Tae Oh
LEMMA 1.
Let 9 be an arbitrary left invariant Riemannian metric on M. Let ( , ) be an inner product on g defined by (X, Y) := ge(Xe,Ye), where X, YE g and e is the identity matrix of M. Then there exists an orthonormal basis (Xh X 2'X
3 )of g with respect to (')0 such that
{
[XI, X
2 ]= (1/.../2)X
3 ,[X
3 ,Xl] = (1/../2)X
2 ,[X
2 ,X
3 ]= (l/V2)Xh
(Xi, Xj) = Dij a~, (2)
where ai,(i = 1,2,3), are positive constant real numbers determined by the given left invariant Riemannian metric 9 of M.
We fix an orthonormal basis (XI. X
2 ,X
3 )of g with respect to go with the property (2) in Lemma 1 and denote by g(aha2,a3), or simply by g( a), the left invariant Riemannian metric on M which is determined by positive l'ealnwnbel's
81,a2,o,3in Lemma 1.
The connection function
aon g
Xg corresponding to the left invariant Riemannian connection of (M, g) is given as follows (cf. [2, p. 52]):
{
a(X, Y) = 1/2 [X, Y] + U(X, Y), (X, Y) : = ge(X
e , ~),where U(X, Y) is determined by
(X,Y E g),
(3)
2(U(X, Y), Z) (4)
=([Z, X], Y) + (X, [Z, Y]), (X, Y, Z E g).
Using (2) and (4), we
g~t LEMMA2.
I U(Xh U(X2, X 3) U(X U(Xi, X3 , Xl). Xl)
2 ) = U(X
2 , X 2) = U(X3, X 3) - 0,
= (V2/4)aa 2 (ai - al)X
3 ,= (V2/4)a 1 2 (al- a22)X
b= (V2/4)ai 2 (al- al)X2,
(5)
for the orthonormal basis (Xl, X
2 ,X
3 )with respect to ( , )0 and
positive numbers al,a2,a3. in Lemma 1.
Scalar curvatures of invariant metrics
631(Xt/aI,X2/a2,X3/a3) is an orthonormal frame basis of (M,g(a».
Let R (resp. S(g(a») be the llicci tensor (resp. the scalar curvature) of (M,g(a». Then we have from (2), (3), and (5)
where Yi := Xi/ai, i = 1,2,3 and c:= a
12alal. We get from (6)
The first author obtained the folowing result (cf. [3, Proposition 3.3]).
LEMMA 3. An inner automorphism A x, (x = exp(rXI), r ER), of (M, g(a» is harmonic if and only if
a2
=
a3or rE { nrr/V2 I n is an integer}. (8)
Thus we get from (7) and Lemma 3
THEOREM 1. Let A x, (x = exp(rX
1 ),r ER), be an inner auto- morphism of (M, g(a». Assume that a
12= 2(a
22+ a3 2) and the scalar curvature of (M, g( a) is zero. Then A x is a harmonic map.
THEOREM 2. AssumeinnerautomorphismsAx,(x = exp(rX
1 ),r
~{nrr /..;2 In is an integer }), of (M,g(a») are harmonic maps. Then, the scalar curvature S(g(a» of (M,g(a») is as follows:
In the cases of
a12~ 4 al,
a 12= 4 al and
a 12~ 4 a2
2,S(g(a»
is positive, zero and negative respectively.
REMARK. Similarly, the Theorems above can be stated for inner
automorphisms A x of (SU(2),g(a», (x = exp(rX2) or exp(rX3»,
(cf. [3, Proposition 3.4 and 3.5]).
632 Joon-Sik Park and Won-Tae Oh
References
1.
