On rigidity phenomena of compact surfaces in homogeneous $3$-manifolds

Abstract

© 2015 American Mathematical Society Let E(κ, τ) be the 3-dimensional homogeneous Riemannian manifold with isometry group of dimension 4, where κ is the curvature of the basis and τ the bundle curvature, which satisfy κ − 4τ 2 ≠ 0. A special case of E(κ, τ) is the Berger sphere that is also denoted by S 3 < inf > b (κ, τ). In this paper, surfaces of E(κ, τ) are studied. As the main result, rigidity theorems in terms of the second fundamental form are established for compact (minimal) surfaces of S 3 < inf > b (κ, τ).