Abstract
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 β p (M) = 0 for 2 ≤ p ≤ 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 ≤ p ≤ n−2.
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Lin, H. On the Structure of Submanifolds in Euclidean Space with Flat Normal Bundle. Results. Math. 68, 313–329 (2015). https://doi.org/10.1007/s00025-015-0435-5
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DOI: https://doi.org/10.1007/s00025-015-0435-5