- CONNECTION, CURVATURE, AND COVARIANT DIFFERENTIATION 63
SOLUTION. We computer^1 &/3 --- ~ 29 18 (~ f)za9f38 - + f)z/39a8 ~ - - f)z89af3 ~ -) -- g^18 f)za9f38 ~ -
= g'Y8-g -^8 - = -r'Y
f)za /38 af3·
Similarly to grc and Ve, we may define Rmc and Rec as the complex
multilinear extensions of Rm and Re. The components of the curvature
(3, 1)-tensor Rmc are defined by(8 8)8· 8 8 J 8
Rm re f)za' az/3 az'Y =;= RO!Jh az8 + Ra"/J'Y az&'(8 8)8· 8 8 J 8
Rm re f)za' az/3 8z1 =;= Ra"/31 az8 + Ra"/31 az&'etc. As a ( 4, 0)-tensor the components of Rm re, which are defined byRa"/J'YJ ~ Rm re (()~a, f)~/3, f)~'Y, ()~8) 'satisfy
etc.
The only nonvanishing components of the curvature (3, 1)-tensor are
R~"!J'Y, R!"/3 1 , R~/3'Y, R~ 131.In particular, R~/3'Y = R~ 131 = 0, etc. Hence, the only nonvanishing compo-
nents of the curvature (4, 0)-tensor are
Ra"!J'YJ, Ra"/3 1 8, Raf3'YJ, Ra/318.Since a~°' I'$'Y = 0, we have
) R8 _ ~r8 _ ~ -~ 8r; 8fi^8
2
_
(^2 .4 Ol./3"( - az/3 OI."( - f)za^9 'Y'f/ f)z/3^9 g f)z0i.f)zf3^9 'Y'f/'and thus we have the following lemma.
LEMMA 2.10 (Kahler Rm). In holomorphic coordinates, the components
of Rmc are given by
82 A- a a(2.5) Ra"!J'YJ = -azaf)z/39'YJ + 9 μ, f)zag'Yμ 8213 9>.J·
We have the identitiesROl."/J'YJ = R'Y"!JaJ = RaJ"("(J = R'YJa"(J
and
Ra"!J'YJ = R13a81 ·
The vanishing of some of the components of the curvature ( 4, 0)-tensor
is related to the following.