A note on the dimension of isometry group of a Riemannian manifold

NTMSCI 5, No. 2, 273-276 (2017) 273 New Trends in Mathematical Sciences http://dx.doi.org/10.20852/ntmsci.2017.177 A note on the dimension of isometry group of a Riemannian manifold Hulya Kadioglu Yildiz Technical University, Istanbul, Turkey Received: 9 August 2016, Accepted: 27 February 2017 Published online: 12 August 2017. Abstract: In this paper, we obtain some results on the dimension of the isometry group of a Riemannian manifold. In specific dimensions, we give a range which the dimension of an isometry group can not be in. We also give necessary conditions for a manifold to have a free canonical action on some specific manifolds. We give a boundary of the dimension of the full isometry group if the dimension of a manifold is greater or equal to 4. Keywords: Isometry groups of riemannian manifolds, noncompact lie groups of transformations, manifolds of metrics. 1 Introduction It is well known that the group of isometries of a Riemannian manifold carries a Lie group structure acting on M as a Lie transformation group. Therefore an isometry group of a Riemannian manifold has both algebraic and smooth structures. Its manifold topology is the compact-open topology which was introduced by Ralph Fox in 1945 [2]. This group is compact if the manifold is compact. On the other hand the boundaries of the dimension of isometry groups has long been studied by researchers. If M is an n− dimensional Riemannian manifold and G is a closed subgroup of I(M), the group of isometries of M, it is a classical result that dimG < 21 n(n + 1). H. C. Wang [11], and H. Wakakuwa [10] gives some other results on the dimension of the isometry group. Moreover Ihrig [3] proved that under suitable smoothness conditions, the upper bound is n(n+1) if M is 2 an n−dimensional manifold which is homeomorphic to a topological metric space. In this paper, we give some other boundaries if the manifold M or its canonical action carries certain conditions. If Dim(M) ≥ 4, then the upper bound is n(n−1) + 1 if M is not of constant curvature. We also prove that if M is an 2 n−dimensional Riemannian manifold, where M is isometric to one of the elements of A = {Sn, Rn , Pn (R), Hn } of constant curvature, and (E, π , M, E) be a vector bundle, then if the canonical isometric action on M is free , there is a following relation between Dim(M) = n and Dim(E) = k, k= t(t − 1) ⇐⇒ n ∈ {1, 2, ...,t} 2 (1) where t = 2, 3, ..... In this study, we assume that all manifolds are smooth, second countable, and Hausdorff. ∗ Corresponding author e-mail: c 2017 BISKA Bilisim Technology 274 H. Kadioglu: A note on the dimension of isometry group of a Riemannian manifold 2 Preliminaries In this section, we give some preliminary information that we use throughout this paper. First we define induced metric on the total space of a vector bundle. Let π : E → M be a vector bundle, where (M, g) is an n-dimensional Riemannian manifold, and suppose that (U, E, Φ ) is a local fiber bundle trivialization of E with h ∈ π −1 (U) and identify T E with E × E. It was proven in [4] that there exists an induced Riemannian metric e g on E as follows. g̃(Vh ,Wh ) = g((π )∗ (V ), (π )∗ (W )) + Q((pr2 ◦ Φ )∗ (V ), (pr2 ◦ Φ )∗ (W )) (2) where Q((e, v), (e′ , v′ )) =< e, e′ > + < v, v′ > for all (e, v), (e′ , v′ ) ∈ T E. In [5], it was proven in [5] that using the above metric, one can define a one to one Lie group homomorphism between isometry groups of base and total space of the vector bundle. The following theorem takes care of this issue. Theorem 1. [5] Let I(M) and I(E) denote isometry groups of M and E respectively. Suppose that for each x ∈ M we fix a local trivialization Φ around x, and denote it as Φx (similarly, when we pick another point y ∈ M, the fixed trivialization will be Φy ). Therefore when we pick a point x ∈ M, then Φx is uniquely defined. Let f ∈ I(M). Then the function Ω : I(M) → I(E) f→ Ω ( f ) = Φ −1 f (x) ( f ◦ π , pr2 ◦ Φx ) (3) is a Lie group homomorphism. It was also proven that the image of this homomorphism is a closed subgroup of the isometry group of E. Now, let e G = I(M) and the image Im(Ω ) = G. Theorem 2. [5] Let G̃ denotes the image of the function Ω . Then e = {F ∈ I(E)| F = Φ −1 ( f ◦ π , pr2 ◦ Φx ), f ∈ G} G f (x) (4) is a Lie subgroup of I(E). e carries the relative topology to I(E). This makes the restriction of the function Ω ( the restriction to its The subgroup G image) a Lie group isomorphism. e Corollary 1.[5] The function Ω |Ω −1 (G) e is a Lie group isomorphism. Thus G is of dimension N, where Dim(I(M)) = N. We use the above corollary to decide the dimension of the isometry group of M in certain conditions. Theorem 3. [8] Let M be an n-dimensional Riemannian manifold with n 6= 4, 6, 10. Then the group I(M) of isometries contains no closed subgroup G where the dimension of G falls into any of the ranges: 1 1 1 (n − k)(n − k + 1) + k(k + 1) < dimG < (n − k + 1)(n − k + 2), k = 1, 2, 3, ... 2 2 2 (5) 3 Theory n(n+1) It is well known that the dimension of an isometry group of a Riemannian manifold is at most 2 . In this section, we give some boundaries to the dimension of isometry groups by using the structures defined in previous sections. First we define the induced action on the total space of a vector bundle. c 2017 BISKA Bilisim Technology NTMSCI 5, No. 2, 273-276 (2017) / www.ntmsci.com 275 Definition 1. Let µ be the canonical action of I(M) on M. Then the canonical G̃ action on E is defined in a usual way as follows. µ̃ (F, h) = F(h), (6) e We will call this action as the induced action on E. where F ∈ G. Theorem 4. Let (E, π , M) is a vector bundle, in which the total space of the bundle endows with the induced metric gE . Then the following properties hold e acts freely on E. (i) If I(M) acts freely on M, then G (ii) If dim(E) 6= 4, 6, 10, then the dimension of isometry group I(M) does not fall into any of the ranges 1 1 1 (n − k)(n − k + 1) + k(k + 1) < dim(I(M)) < (n − k + 1)(n − k + 2), k = 1, 2, 3, ... 2 2 2 (7) e is an imbedded Lie subgroup of I(E). Then from Proposition 3.62 in [1], the induced action (i) By Theorem 2, G e e From µ̃ : G × E → E is proper. On the other hand, let F(h) = h for some h ∈ π −1 {x} for some x ∈ M, where F ∈ G. Equation 4, Φ f (x) (h) = (( f ◦ π )(h), (pr2 ◦ Φx )(h)), Proof. which implies )(( f ◦ π )(h), (pr2 ◦ Φ )(h)) = pr1 (( f ◦ π )(h), (pr2 ◦ Φ )(h)) = ( f ◦ π )(h). π (h) = π (F(h)) = (π ◦ Φ −1 f (x) (8) Since µ is a free action, then f = idI(M) . Because Ω is a homomorphism, then it maps identity of I(M) to identity e F(h) = h implies F = id, then µ̃ is a free of I(E), which implies that F = idGe . Since for any h ∈ E, and F ∈ G, action. (ii) We recall that Mann [8] has proven that if an n−dimensional Riemannian manifold M with n 6= 4, 6, 10, then I(M) contains no closed subgroup G where dimension of G falls into any of the ranges in Equation 7. Since (E, gE ) is e is a closed subgroup of a Riemannian manifold (endowed with (...truncated)