Isometry groups of orthonormal frame bundles

✍ Scribed by Robert J. Fisher; Kyung Bailee

• Publisher: Springer
• Year: 1986
• Tongue: English
• Weight: 258 KB
• Volume: 21
• Category: Article
• ISSN: 0046-5755
• DOI: 10.1007/bf00182905

✦ Synopsis

Let M be a connected, oriented Riemannian n-manifold, and let SO(M) be its bundle of oriented orthonormal frames. This principal SO(n)-bundle SO(M)~,M will have a Riemannian metric coming from the Levi-Civita connection together with the bi-invariant metric on SO(n), namely, -1/[2(n-1)] times the Killing form. The space SO(M) with this metric described above can be useful. For example, if M is the 2-dimensional hyperbolic space H 2, then SO(M) is the unit tangent bundle. The main concern of this paper is to study the group IF(SO(M)) of fiber-preserving isometries of SO(M). To describe the results, let I(M) denote the group of isometries of M, O(u) the homogeneous holonomy of M thru a frame u ~ SO(M), and let C(u) be its centralizer in SO(n). The subscript o of any Lie group means its connected component. For a group G, l(G) and r(G) denote the left and right translations, respectively. Our main results are A. There is an isometric action of C(u) on SO(M) as 'left translations' on each fiber.

B. I~(SO(M)) is naturally isomorphic to (G(u). SO(n)) o Io(M ).

C. The centralizer C(u) of the holonomy 9roup C~(u) has at most two components and when n is odd it is connected. D. The space SO(M) contains a subbundle which is isometric to the Lie 9roup lo(M) with a natural left invariant metric whenever M is a homogeneous Riemannian manifold.