- CONNECTION, CURVATURE, AND COVARIANT DIFFERENTIATION 67
Let V '13 ~ V 81 ozf3. The Laplacian acting on tensors is given by
1 - 1
(2.13) ~ ~ 2l^43 (VaVfi + VfiVa) = 2 (Va Va+ Va Va).NOTATION 2.17. At this point we have begun to use the extended Ein-stein summation convention where not only pairs of repeated indices
consisting of an upper index and a lower index are summed but also pairs
of repeated lower indices consisting of a barred index and an unbarred index
are summed. For example, aaa ~ 'L:~,,B=I gafiaa'/3· Formulas which are ex-
pressed using this convention hold in the literal sense in unitary holomorphic
coordinates at a point p. Usually we use the extended Einstein summation
convention in lieu of computing in unitary coordinates at a point. In this
sense, the formulas in (2. 7) and throughout this chapter hold in arbitrary
holomorphic coordinates.
EXERCISE 2.18. Show that the Laplacian defined above in (2.13) is one-
half of the Riemannian Laplacian.Acting on functions, the Laplacian isA a'/Jn n ~ a'/3 °
2
L..l - g v a v ,B - g f}zCY.f}z,B 'since, acting on functions, Va V '13 = oz~~zf:! = V '13 Va· We also note that if </>is a real-valued function on M, then
(2.14) ~ lgrad</>1^2 = gafiVa</>V'jJ</> ~ IV</>1^2 ,where the LHS is the Riemannian gradient. Likewise,
1 2 I
1(2.15) 2 IHess </>I = Va V '13</>^2 + IV a V ,e</>I^2 ,
where the LHS is the Riemannian Hessian.
In our calculation of evolution equations under the Kahler-R!cci fl.ow we
shall often use the following.
LEMMA 2.19 (Kahler commutator formulas). On a Kahler manifold we
have the following commutator formulas for covariant differentiation acting
on tensors of type (1, 0), (0, 1), and (1, 1), respectively:
(2.16)(2.17)
(2.18)Va V '/Ja'Y - V '/JV aa'Y = -R~fi'Yao,VaV'13b1 - VfiVab1 = -R~'/3 1 b5 = Ria'Yb5,
Va V fia'Y5 - V '13 V aa'Y5 = -Rafi'Yfia'T/5 + Ra'/3'TJ8a'Yfi.
Analogous formulas hold for higher degree tensors and forms (see Exercise
2.22).REMARK 2.20. Note that Ra'/J'T/8 = -Ra'/38iJ? which exhibits the con-
sistency of the formula above with the analogous formula in Riemannian
geometry. We have also used the extended Einstein summation convention.