On the Structure of Submanifolds in Euclidean Space with Flat Normal Bundle

Abstract

In this paper we study the structure of an immersed submanifold Mn in a Riemannian manifold with flat normal bundle in two ways. Firstly, we prove that if Mn is compact and satisfies some pointwise pinching condition, and assume further that the ambient space has pure curvature tensor and non-negative isotropic curvature, then the Betti numbers βp(M) = 0 for 2 ≤ p ≤ n−2. Secondly, suppose that Mn is a complete non-compact submanifold in the Euclidean space with finite total curvature in the sense that its traceless second fundament form has finite Ln-norm, then we show that the spaces of L2 harmonic p-forms on Mn have finite dimensions for all 2 ≤ p ≤ n−2.