ON THE HOLONOMY GROUP OF HYPERSURFACES OF SPACES OF CONSTANT CURVATURE 5 OGNIAN KASSABOV 1 0 2 Abstract. WeclassifyhypersurfacesMn ofmanifoldsofconstantnonzerosectionalcurva- b ture according their restricted homogeneous holonomy groups. It turns out that outside of e the evident cases (restricted holonomy group SO(n) and flat submanifolds) only two cases F arise: restricted holonomy group SO(k)×SO(n−k) (when M is locally a product of two 0 space forms) and SO(n−1) (when M is locally a product of an (n−1)-dimensional space 2 form and a segment). ] G D . 1. Introduction h t a The holonomy groups are fundamental analytical objects in the theory of manifolds and m especially in the theory of Riemannian manifolds. The holonomy group of a Riemannian [ manifold reflects for example on local reducibility of the manifold. In [6] M. Kurita classifies 2 the conformal flat Riemannian manifolds according their restricted homogeneous holonomy v group. 7 2 There exists a similarity between the conformal flat Riemannian manifolds and the hyper- 6 surfaces of a Riemannian manifold, see e.g. a remark of R. S. Kulkarni in [5]. So it is natural 6 to look for a result in the submanifold geometry, analogous to the Kurita’s theorem. In [3] . 1 S. Kobayashi proves that the holonomy group of a compact hypersurface of En+1 is SO(n). 0 4 Generalizations of of Kobayashi’s result are obtained by R. Bishop [1] and G. Vranceanu [8]. 1 In this paper we consider analogous question for hypersurfaces of non-flat real space forms : v according their holonomy groups. Namely we prove: i X Theorem 1. Let Mn (n ≥ 3) be a connected hypersurface of a space Mn+1(ν) of constant r a positive sectional curvature ν. Then the restricted homogeneous holonofmy group Hp of Mn in any point p is in general the special orthogonal group SO(n). If H is not SO(n) at any p point p ∈ Mn, then one of the following cases appears: a) H = SO(k)×SO(n−k), 1 < k < n−1 and Mn is locally a product of a k-dimensional p space of constant curvature ν + λ2 and an (n − k)-dimensional space of constant sectional curvature ν +µ2, with ν +λµ = 0; b) H = SO(n−1) and Mn is locally a product of an (n−1)-dimensional space of constant p sectional curvature and a segment. A similar theorem for complex manifolds is proved in [7]. 2. Preliminaries. Let Mn+1 be an(n+1)-dimensional Riemannian manifoldwith metric tensor g and denote by ∇ ifts Riemannian connection. It is well known that if Mn+1 is of constant sectional e f Key words and phrases. Space of constant curvature, hypersurface, holonomy group. 2010 Mathematics Subject Classification: 53B25. 1 2 OGNIANKASSABOV curvature ν, then its curvature operator R has the form e R(x,y) = νx∧y , e where the operator ∧ is defined by (x∧y)z = g(y,z)x−g(x,z)y . Such a manifold is denoted by Mn+1(ν). Now let Mn be a hypersurface of Mn+1(ν) and denote by ∇ its Riemannian confnection. Then we have the Gauss formula f ∇ Y = ∇ Y +σ(X,Y) X X for vector fields X,Y on Mn, wheere σ is a normal-bundle-valued symmetric tensor field on Mn, called the second fundamental form of Mn in Mn+1. Let ξ be a unit normal vector field. Then the Weingarten formula is f ∇ ξ = −A X x ξ e and the operator A is related to σ by ξ g(σ(X,Y),ξ) = g(A X,Y) = g(A Y,X) . ξ ξ Suppose that we have fixed a normal vector field ξ. Then we shall write A insteed of A . ξ The equations of Gauss and Codazzi are given respectively by R(X,Y) = ν(X ∧Y)+AX ∧AY , (∇ A)Y = (∇ A)X , X Y R denoting the curvature operator of Mn. It is known that the Lie algebra of the infinitesimal holonomy group at a point p of a Riemannian manifold M is generated by all endomorphisms of the form (∇kR)(X,Y;V ,...,V ) , 1 k where X,Y,V ,...,V ∈ T M and 0 ≤ k < +∞ [4]. Moreover if the dimension of the infin- 1 k p itesimal holonomy group is constant, this group coincides with the restricted homogeneous holonomy group [4]. 3. Proof of Theorem 1. Let p be an arbitrary point of Mn. We choose an orthonormal basis e ,...,e of T M, 1 n p which diagonalize the symmetric operator A, i.e. Ae = λ e i = 1,...,n . i i i Then by the equation of Gauss we obtain (3.1) R(e ,e ) = (ν +λ λ )e ∧e . i j i j i j First we note that Mn cannot be flat at p. Indeed if Mn is flat, we obtain from (3.1) ν +λ λ = 0 for all i 6= j. Since n > 2 this implies easily ν +λ2 = 0, and because of ν > 0 i j 1 this is a contradiction. Since Mn is not flat at p, there exist i 6= j, such that ν+λ λ 6= 0. Then (3.1) implies that i j e ∧e belongs to the Lie algebra h of H . As in [6] we denote by SO[i ,...,i ] the subgroup i j p p 1 k of SO(n), which induces the full rotation of the linear subspace, generated by e ,...,e and i1 ik fixes the remaining vectors. Denote also by so[i ,...,i ] the Lie algebra of SO[i ,...,i ]. Then 1 k 1 k according to the above argument H contains SO[i,j]. p ON THE HOLONOMY GROUP OF HYPERSURFACES 3 If H contains SO(n), then H = SO(n), because the restricted homogeneous holonomy p p group H of a Riemannian manifold is a subgroup of SO(n), see [2]. p Let H is not SO(n). Then there exist k, 2 ≤ k ≤ n−1 and indices i ,...,i , such that H p 1 k p contains SO[i ,...,i ] but doesn’t contain SO[i ,...,i ,u] for u 6= i ,...,i . Without loss of 1 k 1 k 1 k generality we can assume that H contains SO[1,...,k], but does not contain SO[1,...,k,u] p for u > k. Let us suppose that h contains so[a,u] for some a ∈ {1,...,k} and u ∈ {k + 1,...,n}. p Since [e ∧e ,e ∧e ] = e ∧e b a a u b u it follows that the Lie algebra h contains e ∧ e for b = 1,...,k. Hence h contains p b u p so[1,...,k,u], which is a contradiction. Consequently h doesn’t contain so[a,u] for any a = 1,...,k; u = k +1,...,n. Then (3.1) p implies (3.2) ν +λ λ = 0 a = 1,...,k; u = k +1,...,n. a u Hence, using ν 6= 0, we obtain λ = ... = λ and λ = ... = λ . Denote λ = λ ; θ = λ . 1 k k+1 n 1 k+1 Then by (3.2) ν +λθ = 0, λ 6= 0, θ 6= 0 and it follows easily λ 6= θ, ν +λ2 6= 0, ν +θ2 6= 0. In a neighborhood W of p we consider continuous functions Λ ,...,Λ , such that for any 1 n point q ∈ W the numbers Λ (q),...,Λ (q) are the eigenvalues of A. Since ν + λ2 6= 0, 1 n ν +θ2 6= 0, then in an open subset V of W containing p we have ν +Λ (q)Λ (q) 6= 0 a,b = 1,...,k ; a b ν +Λ (q)Λ (q) 6= 0 u,v = k +1,...,n . u v Hence H containsSO[1,...,k]andSO[k+1,...,n]. Supposethatν+Λ (q)Λ (q) 6= 0forsome q a u a = 1,...,k, u = k +1,...,n. Then h contains e ∧e , so as before h contains so[1,...,k,u] q a u q and analogously h contains so(n), which is not possible. So ν +Λ (q)Λ (q) = 0. Hence as q a α before we find Λ (q) = ... = Λ (q) , Λ (q) = ... = Λ (q) . 1 k k+1 n Consequently in a neighborhood V of p there exist a number k and continuous functions Λ(q),Θ(q) such that Λ(q) 6= Θ(q) and (3.3) Λ (q) = ... = Λ (q) = Λ(q) 6= 0 , Λ (q) = ... = Λ (q) = Θ(q) 6= 0 1 k k+1 n for q ∈ V. Since Mn is connected k is a constant on Mn. Consequently (3.3) holds on Mn. On the other hand using ν +ΛΘ = 0 and the fact that kΛ+(n−k)Θ = trA is smooth we conclude that Λ and Θ are smooth functions on Mn. Define two distributions T (q) = {x ∈ T (M) : Ax = Λ(q)x} , 1 q T (q) = {x ∈ T (M) : Ax = Θ(q)x} . 2 q It follows directly that T and T are orthogonal and for X,Y ∈ T , Z,U ∈ T we have 1 2 1 2 R(X,Y) = (ν +Λ2)X ∧Y , ν R(Z,U) = (ν +Λ2)Z ∧U , Λ2 R(X,Z) = 0 . 4 OGNIANKASSABOV We choose local orthonormal frame fields {E ,...,E } of T and {E ,...,E } of T and we 1 k 1 k+1 n 2 denote n ∇ E = Γ E . Ei j X ijs s s=1 Then Γ = −Γ for all i,j,s = 1,...,n, in particular Γ = 0. As before let a,b,c ∈ ijs isj ijj {1,...,k} and u,v ∈ {k +1,...,n}. From the second Bianchi identity we have (∇ R)(E ,E )+(∇ R)(E ,E )+(∇ R)(E ,E ) = 0 a b u b u a u a b and hence k E (Λ2)E ∧E +(ν +Λ2) {Γ E ∧E −Γ E ∧E } u a b X buc a c auc b c c=1 n ν +(ν +Λ2) (Γ −Γ )E ∧E +Γ E ∧E −Γ E ∧E = 0 . X nΛ2 abv bav u v uav v b ubv v ao v=k+1 Consequently we obtain (3.4) E (Λ2) = (ν +Λ2){Γ +Γ } , u aau bbu (ν +Λ2)Γ = 0 uva for all a 6= b. Since ν +Λ2 6= 0 we find Γ = 0, so T is parallel. uva 2 Let n−k ≥ 2. Then analogously to the above T is also parallel. Now (3.4) implies that 1 Λ doesn’t depend on E and analogously Θ doesn’t depend on E . Hence, using ν+ΛΘ = 0 u a we conclude that Λ and Θ are constants. So we obtain the case a) of our Theorem. Let n−k = 1. We shall show that under the assumption H 6= SO(n) the distribution T p 1 is again parallel. By the Codazzi equation we have (∇ A)(E ) = (∇ A)(E ) . a b b a This implies E (Λ)E +(Λ−Θ)Γ E = E (Λ)E +(Λ−Θ)Γ E . a b abn n b a ban n Hence E (Λ) = 0 for a = 1,...n−1. Now from a (∇ A)(E ) = (∇ A)(E ) a n n a we obtain n−1 E (Λ)E +(Λ−Θ) Γ E = 0 . n a X anc c c=1 Hence we derive (3.5) E (Λ) = (Λ−Θ)Γ , n aan (Λ−Θ)Γ = 0 for c 6= a . acn Since Λ 6= Θ the last equality implies Γ = 0 for a 6= c. On the other hand (3.5) implies acn Γ = Γ . If Γ = 0, then T is parallel and from (3.5) E (Λ) = 0, so Λ is a constant. aan bbn aan 1 n Because of ν +ΛΘ 6= 0 it follows that Θ is a constant too. Hence we obtain the case b) of our Theorem. Let us suppose that Γ 6= 0. We compute directly aan (∇ R)(E ,E ) = (ν +Λ2)Γ E ∧E . a a b aan n b Hence E ∧E ∈ h and as before it follows that SO(n) = H , which is not our case. This n b p p proves Theorem 1. ON THE HOLONOMY GROUP OF HYPERSURFACES 5 Remark. Inthesameway we canconsider thecasewhere Mn+1(ν) isofconstant negative sectional curvature ν. Then we obtain f Theorem 2. Let Mn (n ≥ 3) be a connected hypersurface of a space Mn+1(ν) of constant negative sectional curvature ν. Then the restricted homogeneous holonofmy group H of Mn p in any point p is in general the special orthogonal group SO(n). If Mn is not flat and H is p not SO(n) at any point p ∈ Mn, then one of the following cases appears: a) H = SO(k)×SO(n−k), 1 < k < n−1 and M is locally a product of a k-dimensional p space of constant curvature ν + λ2 and an (n − k)-dimensional space of constant sectional curvature ν +µ2, with ν +λµ = 0 b) H = SO(n−1) and M is locally a product of an (n−1)-dimensional space of constant p sectional curvature and a segment. References [1] R.Bishop,Theholonomyalgebraofimmersedmanifoldsofcodimensiontwo,JournalofDiffer.Geometry 2(1968), 347-353. [2] A. Borel and A. Lichnerowicz, Groups d’holonomie des vari´et´es riemanniennes, C. R. Acad. Sci. Paris 234(1952), 1835-1837. [3] S. Kobayashi,Holonomy group of hypersurfaces, Nagoya Math. Journal 10(1956),9-14. [4] S. Kobayashi and K. Nomizu, Foundations of differential geometry, Vol. I, John Wiley and Sons, New York, 1963. [5] R. S. Kulkarni, Equivalence of K¨ahler manifolds and other equivalence problems, Journal of Differ. Geometry 9(1974), 401-408. [6] M. Kurita,On the holonomygroupof the conformallyflat Riemannianmanifold. NagoyaMath. Journal 9(1955), 161-171. [7] K. Nomizu and B. Smyth, Differential geometry of complex hypersurfaces II, J. Math. Soc. Japan 20(1968), 498-521. [8] G. Vranceanu, Sur les groupes d’holonomie des espaces Vn plong´es dans En+p sans torsion, Revue Roumaine de Math. Pures et Appl. 19(1974), 125-128. Department of Mathematics and Informatics University of Transport 158 Geo Milev Str. 1574 Sofia, Bulgaria E-mail address: [email protected]