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

In this paper we study the structure of an immersed submanifold M (n) in a Riemannian manifold with flat normal bundle in two ways. Firstly, we prove that if M (n) 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 beta (p) (M) = 0 for 2 a parts per thousand currency sign p a parts per thousand currency sign n-2. Secondly, suppose that M (n) 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 L (n) -norm, then we show that the spaces of L (2) harmonic p-forms on M (n) have finite dimensions for all 2 a parts per thousand currency sign p a parts per thousand currency sign n-2.