Compact homogeneous hypersurfaces

Abstract

By homogeneous we mean that the group I(M) of isometries of M is transitive on M. By imbedding we mean a locally one-to-one mapping of M into Rn+l of class C2. The outline of the proof is as follows. (1) Since M is compact, there exists a point of M in a neighborhood of which M is locally convex. (2) This neighborhood is rigid if n > 3. (3) Using the homogeneity of M and the rigidity of the above neighborhood, we show that M is orientable and rigid. (4) Furthermore, the spherical map of Gauss is a homeomorphism of Ml onto the unit sphere. (5) Since M is rigid, every isometry is the restriction of a motion in Rn+1 to M. (6) Therefore I(M) can be considered as a subgroup of the group of motions in R . Since I(M) is compact, it has a fixed point in R . (7) M is a sphere with its center at this fixed point. (8) The case n =2 is treated at the end of the paper. In the last section we indicate other proofs of the theorem which makes use of the theory of convex hypersurfaces. For these alternative proofs, the author owes many suggestions to Professors H. Busemann and S. S. Chern. In the first two sections, we give, for the sake of completeness, a proof of the classical rigidity theorem which plays an important role in this paper. Although we make use of the connection theory in fibre bundles, our proof of the rigidity theorem is not new but due to E. Cartan. 1. Preliminary propositions. NOTATIONS. We denote by Rn the n-dimensional real vector space (with or without a positive definite inner product) or the n-dimensional Euclidean space. If M11 is a manifold T.(M) denotes the tangent space to M at a point x. Let f be a mapping of a manifold M' into M. Then 8f: T(M')->T(M) is the differential of f, where T(M) =Ux Tx(M). If co is a differential form on M, then f* (co) is the differential form on M' induced from co by f.