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(2) Vol. 22, No. 1 Journal of Hokkaido University of Education (Section II A) September 1971. A Note on the Sum of Connections. Masaki FUKUI The Department of Mathematics, Asahikawa Branch, Hokkaido University of Education. »^N: ^o^il-t-6—^t »at^rimjii^?^ai Summary Let co and w' be two connection forms on a principal bundle P, f a C°° function on the base manifold At, and TF the projection from P to M. Then it is known that <y/= (/7r)<y+ (1— /7r)<y/ is also a connection form on P and so is Wa=aci)+ (l—ff)&)', where ff is constant. We. will investigate the relation between the two old connections and the new one, in particular, between their curvature forms. We refer the reader to Kobayashi and Nomizu [2~\ for the notations used in this paper.. We begin with the lemma which is an answer to Problem 2 of Chapter 5 of Bishop and. Crittenden [1J. Lemma 1. If X^Tu(P') and X=hX+Xi, X=h'X+X[ are the horizontal and vertical decompositions of X with respect to w and w', then X is decomposed zuith respect to co/ as. follozus :. X={WhX+a-f7c)h'X}+{WX,+^-f7t~)X[} Proof. It suffices to prove that the first bracket is horizontal with respect to a>/. Their exist such Ai and A[ eg that Xi=Ai* and ^=A^* at u^P. This gives Ai=w(/?/X) +A[, A[=w'(,hX')+Ai, and therefore. w{h'X)+w'(hX)=0. From this and by the definition of &»/, we have. Mf{WkX+ (1-/7T)/Z/X}=0. In particular, the decomposition of X with respect to a)a, is. X={ahX+a-a)h'X}+{aX,+(l-a)X[}. Notice that the horizontal component with respect to cDa is also- written in the form of hX+. (l-ff)(Zi-ZO or h'X+a(X[-X,). Theorem 1. Let Q, Q' and Qa. be the cztrvature forms of co, w' and wa respectively.. Then ^a=a.^+ (l-a)^'-ia(l-a)\_-c, r],. tuhere T=W—W', which is called "the difference form." Proof. This theorem is immediately derived from the structural equations of the three connections.. (4).
(3) ' 22 ^ ^ 1 ^ WMW^CT (^ 11 ^ A) TO 46 ^ 9 H fia,=dWa,+^[_a)a, (Da]. =a{Q-^w, o)]) + (1-ff) (J2/-i[a)/, co']) +i[fffl)+ (l-fi)^/, ^+ (l-ff)a)'] =aQ^-(l-a)Q'-ia(l-a)^(o-w',w-wl~}. It is clear that the above theorem holds for the connection ft)/ with a little modification, that is, if Qf is the curvature form of <y/, then. ^= (/7T)^+(l-/7r)^/-K/7T) (l-/7T)[r, r]+fi?(/7T) AT. Lemma 2. dQ^X, Y, Z)=^C{^{X, Y), <o(Z)J}, d[r, T~](X, Y, Z) =^C{[T-(^), T(r), (y(Z)]}+2)Cr, T'](X, Y, Z) /or X, Y and Z^Tu(,P), ivhere ive denote by C the cyclic sum of X, Y and Z. Proof. If ^ is a g-valued horizontal 2-form of type adG and D as the exterior covariant differentiation with respect to <y, then the following formula is found (see p. 86 of C1J) :. d^(X,Y,Z)=iC{^^X,Y),a(Z^}+D</>(X,Y,Z) To verify the two equations we have only to take Q or [v, r~] as (/), and apply Bianchi's identity. Now we apply the exterior covariant differentiation Da with respect to a)a to the formula in Theorem 1. By Bianchi's identity we have aDaQ + (1 -a) A^/- la (1 -a) £>a[T, r] =0 Noticing the remark which is subsequent to Lemma 1, for X, Y and Z eTu{P). DM{X, Y, Z) =dQ{hX+ (1-ff) (X,-X[), hY+ (1-c) (Vi-yO, hZ+{l-a}{Z,-Z[-)} = (1 - a) {dQ (X, - X[, hY, hZ) + dQ {hX, Y, - Y[, hZ) +dQ(hX,hY,Z,-Z[-)}. =i(l-fl)C{^(^,r),r(Z)]} because dQ vanishes by Lemma 2 if two are vertical and by Blanches identity if all are horizontal with respect to <y. And similarly we have. DaQ'=-^aC{^\X, F),r(Z)]. Therefore, when a is neither 0 nor 1, we have. D^-c, r](X, Y, Z) =|C{[(^-^) (X, D, T(Z)]}. This formula means that the exterior covariant derivative of [r, T'] with respect to a)a is independent of the value of a except 0 and 1. But the exception can be removed by the following lemma. Lemma 3. A([T, f~]=Di^r, T'] for every a where D^ is the exterior covariant cliff erentiation with respect to CD. Proof. For X, Y and ZeT.(P),. Dalr, 0(X, Y, Z) =^[r, r]{/^+ (1-^) ?-^), hY+ (1-fl) {Y,-Y[}, hZ+ (1-fl) (Zi-ZQ}. Here, notice that d[r, T~]{hX, hY, hZ) ==D^T, r](^, Y, Z), d\_r, v'}{hX, hY, Z,-Z[) =i{[r(^), T(^)]» T(-^)) and so on- As other kinds of terms vanish by Lemma 2, we have. Da^r, T^X, Y, Z)=?, T](.Y, Y, Z)+i(l-ff)C{[^(^), r(K):], r(Z)]} =D^r,r-](X,Y,Z) because the second term vanishes by Jacobi's identity.. (5).
(4) Vol. 22, No. 1 Journal of Hokkaido University of Epucation (Section II A) September 1971 By Lemma 3 and the preceding argument, we have the following conclusion. Theorem 2. For every a and X, Y and ZE Tv, (P). D^r, T^X, Y, Z) =|C{[(^-^) (X, V),r(Z)]. REFERENCES [1] Bishop, R. and Crittenden, R. (1964), Geometry oj IVIanifolds. Academic Press, New York and London.. [2] Kobayashi, S. and Nomizu, K. (1963), Foundations of Differential Geometry. Vol. I. Wiley (Interscience). New York.. (6).
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