A Condition for Positivity of Curvature

Introduction The question of positivity of curvature on bundles arose in the middle 60's with O'Neills theorem [O], according to which riemannian submersions increase the sectional curvature. In the early 70's Cheeger and Gromoll in their notable paper [C-G], showed that if an open manifold has a complete riemannian metric with non-negative sectional curvature then it is the total space of a vector bundle over a compact totally geodesic submanifold. Their result is actually quite stronger. A natural question [Yau, problem nO 6] is to ask whether the total spaces of bundles over compact manifolds with non-negative sectional curvature also admit complete metrics with the same property. The approach to this question splits naturally in the cases of principal and vector bundles. Although the vector bundle case can be treated independently IS-W], positive results follow from the principal bundle case through a simple application of O'Neill's theorem. See for example [G-M], [Ch], [R1], JR2]. This note, minus the application, was written in 1979 as an addendum to [D-R] and was never submitted for publication. Due to some revival of interest in the existence of metrics of non negative sectional curvature on vector bundles in recent years ([G], [W2], [S-W]) we thought its publication might be of some help. We remark that the theorem was also known to L. Berard Bergery, at least in Received 8 April 1992, 154 LUCAS M. CHAVES, A. DERDZINSKI AND A. RIGAS the case of G = S 1. Strake and Walschap in [S-W] obtained a sufficient condition similar to (ii) for the non negativity of sectional curvatures in vector bundles. Main Result Let P ~r G be a principal Gbundle, G a compact Lie Group, over a compact manifold M of dimension n _> 2. Given a connection form • in P, a metric h in M, and a bi-invariant metric (,) in G, one can define a family of metrics gt, for t > 0, in P by g t ( X , Y ) = h(Tr , (X) , zr , (Y) ) + t ( w ( X ) , w ( Y ) ) (see [J]), and ask when there are metrics of positive sectional curvature among the gt. With respect to each of these metrics, G acts on P by isometries, the fibers are totally geodesic and therefore by a theorem of Wallach [W1] we have that G = S 1, G = S 3 or G = SO(3) . Our objective here is to find a necessary and sufficient condition so that (P, gt) has positive sectional curvature for small enough t. This is natural since for small t fibers we have "more curvature" and the curvature of the total space tends to increase. Let A d P = P • G be the adjoint fiber bundle of P. The metric (,) of A d induces a metric in the fibers of A d P that we will also denote by (,). I f [2 is the curvature form of w we can consider f2 as a form in M with values in A d P . Now, we can state our result. Theorem. Let P ,r ~. M be a principal G-bundle with a connection ~ over a compact manifold M, d i m M > 2, where G = S ~, SO(3) or S t . Fix a Riemannian metric h in M and a bi-invariant metric (,) in G. Then the following conditions are equivalent: i) There is a to > O, such that for 0 < t < to (P, gt) has positive sectional curvature. ii) For any point z in M, mutually orthonormal unit vectors X , Y in T z M and any non-zero element u of A d ( P ) over z, we have h n R ( X , Y , X , Y ) ~ ( ( u , a ( X , Xk) ) ) ~ > ( ( u , ( V x f ~ ) ( X , Y ) ) ) = k = l h where R is the curvature tensor of (M, h), ~2 is the curvature form oflw, v Bol. Soc. Brat. Mat., Vol. 23, Ns. 1--2, 1992 A CONDITION FOR POSITIVITY OF CURVATURE 155 is the covariant derivative o f A d ( P ) induced by w and { X i , . . . , X,~} is an arbitrary orthonormal basis o f T~M. For the calculation of the components of the curvature tensor R t of (P, gt) we refer the reader to Jensen [J]. The following convention for indices will be used: i < _ i , j , k , m < n = d i m M and n + l < a , b , c , d , f < n + 3. Let {X,~+I, X,~+2, Xn+3} be an orthonormal (o.n.) frame field of left invariant vector fields on G with respect to (,), i.e., an o.n. frame in the Lie algebra of G, taken to be S 3 or SO(3) henceforth. Set [X,~,Xb] = ~ c C ~ X c . Since (,) is bi-invariant, the structure constants {CaCb} are skew-symmetric in every pair of indices. Let w = ~ w'~X,~ where the w" are 1-forms on P and observe that from the right action of G on P, the vectors {X,,} induce three fundamental vertical vector fields ([K-N p. 51]) X* * * n + l , X n + 2 , X n + 3 o n P, which are linearly independent at every point. Relative to the 9t-metric the fields fffa :t i / 2 x ~ n + l < a < n + 3 form an orthonormal vertical frame. Let { X i , . . . , X , } be an h o.n. frame field on an open U __. M. We use the same notation for its lifting to a gt o.n.,w-horizontal frame field on _c M. Define the following functions o n 7r-l(U) H ~ = H,~;~ = <(Vx~ a ) ( X ~ , X j ) , X ~ > R i ~ = 9 t ( R t ( X i , X j ) S ~ , J Q ) , etc. We have Proposition. ([J].) 1 ) t t _ = R~skm t 2 a ( 2 H ~ H ~ m + H a k H ] m Hr ) R ~ j k m 2) t _ t i / 2 H ~ 3) t ~ R~iab = t E~(H?kH~i HbkH~j) E , H } s C ~ b 4) t 1 c a 5) t Rabic : 0 6) t t -~ ~ t"*a t'~f Rabcd = ~ A..,f "'bf"~ cd Bol. Soc. Bras. Mat., Vol. 23, Ns. 1-2, 1992 156 LUCAS M. CHAVES, A. DERDZLNSKI AND A. RIGAS All components of the curvature tensor can be obtained from these using the identities RABCD = --RBACD = --RASDC, RABOD = RODAS and RABCD + RADBC qRACDB -O. The following Lemma simplifies the calculation of sectional curvatures. Lemma 1. Let a c TpP be a bidimensional plane. There is always an orthonormal basis {X, Y} of a, relative to the metric gl, such that { X = otXi + ~Xa with i C j, a ~ b Y = 7 X i q~ X b or2 + f12 = 72 + 65 = 1 Where { Xi, Xy } is a gl-orthonormal pair of horizontal vectors and {Xa, Xb } a gl-orthonormal pair of vertical vectors. Proof. Let {X, Y} be any gl-orthonormal basis of ~r and X = X H + X V, y = y H + y V the decomposition into w-horizontal and vertical components. If h ( Ir . X H, Ir . Y H) ~ 0 then the new gl-orthonormal basis ) ( = c o s ( e ) x s in(e) r , Y = sin(O)X + eos(O)V with cotan(20):= IIYHII~-IIxHI[2 2h(,r .X n , ,r.Y n) satisfies the condition. [] Observe now that in relation to the metric gt {X, Y } is just an orthogonal basis and for Xa -t-1/2Xa, etc., we have ~ 2 x o I = = 1, X : aXi qfltl/2~(a, y = 7 x i + atx/22b, Ilxll~ = ~2 + t/~2, IIYII~ = 7 = + ta =. For the sectional curvature Kt(a), relative to #t, we have, using the above Lemma and Proposition: 1 2 2 t 2tU2aflTZRtja i K,(cr) ( a s + fl:t)(72 + 8:t) {a 7 R~sq + + 2tl/2a276Rtji b + 2taf175Rtbai + 2taf176Rtj~b qtf12 72 Rtai a q2tz/2 f12 76 Rtabai + 2t3/Z af162 Rtba b Bol. Soc. Bras. Mat., Vol. 23, Ns. 1-2, 1992 A CONDITION FOR POSITWITY OF CURVATURE 157 , 2 a 2 ~ 2 0 t t O t 2 ~ 2 R t b i b } -[-~ I J ~ ~ a b a b ] _ 1 h ( ~ + ~t)( '7~ + 6~t) {a2"7~(R,j,i 3t ~ ( H ~ i ) ~ ) e + 2afl'72tH~,i 2a2"76tH~y,~ + 2afl'Tgt(t ~--~ ~ b 1 H~Coo) k c