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The injectivity radius and the fundamental group of compact homogeneous Riemannian manifolds of positive curvature

In this paper we characterize a homogeneous spherical space form whose fundamental group is a binary dihedral group by means of the injectivity radius of the exponential map. As an application of this result we show that the fundamental group of a non-simply connected homogeneous 1/4-pinched Riemannian manifold is isomorphic to a finitesubgroup of the special unitary group SU(2) of degree 2. Moreover we show that the fundamental group of a non-simply connected homogeneous 1/4-pinched Riemannian manifold whose dimension is of 4j +1 (j > 1) is a cyclic group. As is well known, every nontrivial finitesubgroup of SU(2) is a cyclic, binary dihedral or binary polyhedral group ([21]).Therefore the fundamental group of a non-simply connected homogeneous 1/4-pinched Riemannian manifold isisomorphic to one of the finite groups stated above. Let M be an ra-dimensional (m > 3) complete, connected Riemannian manifold and N an ^-dimensional (0 < n < m ―1) connected, compact submanifold (without boundary) embedded in M. Let ExpN : v(N) ―>M denote the normal exponential map of N. Here v(iV)is the total space of the normal bundle of N in M. In the case where A7"is a point of M, v(N) stands for the tangent space to M at that point. We denote by i(N) the injectivityradius of ExpN. It is defined as the supremum of the set of allr > 0 for which Exp^ : vr(N) ―>・M is an embedding, where vr(iV)is the set of allnormal vectors to N of length less than r. The injectivityradius Inj(M) of M is then defined as Inj(M) = inf{i(x) \x e M}. If M is compact, then Inj(M) is positive. A binary dihedral group Dt is a finitegroup generated by two elements a