Complete submanifolds with relative nullity in space forms
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Complete submanifolds with relative nullity in space forms Samuel Canevari1 · Guilherme Machado de Freitas2 · Felippe Guimarães3 · Fernando Manfio4 · João Paulo dos Santos5 Received: 8 June 2020 / Accepted: 28 September 2020 © Springer Nature B.V. 2020
Abstract We use techniques based on the splitting tensor to explicitly integrate the Codazzi equation along the relative nullity distribution and express the second fundamental form in terms of the Jacobi tensor of the ambient space. This approach allows us to easily recover several important results in the literature on complete submanifolds with relative nullity of the sphere as well as derive new strong consequences in hyperbolic and Euclidean spaces. Among the consequences of our main theorem are results on submanifolds with sufficiently high index of relative nullity, submanifolds with nonpositive extrinsic curvature and submanifolds with integrable relative conullity. We show that no complete submanifold of hyperbolic space with sufficiently high index of relative nullity has extrinsic geometry bounded away from zero. As an application of these results, we derive an interesting corollary for complete submanifolds of hyperbolic space with nonpositive extrinsic curvature and discourse on their relation to Milnor’s conjecture about complete surfaces with second fundamental form bounded away from zero. Finally, we also prove that every complete Euclidean submanifold with integrable relative conullity is a cylinder over the relative conullity. Keywords Complete submanifolds · Relative nullity · Splitting tensor · Milnor’s conjecture Mathematics Subject Classification Primary 53C40 · Secondary 53C42
1 Introduction The index of relative nullity introduced by Chern–Kuiper [5] is a fundamental concept in the theory of isometric immersions. At a point x ∈ M n , the relative nullity subspace Δ(x) of f is the kernel of the second fundamental form of f at x, and the index of relative nullity 𝜈(x) is the dimension of Δ(x) . It is well known that, on each open subset where the index The third author was supported by CAPES—Finance Code 001, during his stay at Universidade de Brasília, and by FAPESP Grant #2019/19494-0. The fourth author was supported by FAPESP Grant 2016/23746-6. The fifth author was supported by FAPDF 0193.001346/2016. * João Paulo dos Santos [email protected] Extended author information available on the last page of the article
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Annals of Global Analysis and Geometry
of relative nullity is a positive constant, the submanifold is foliated by totally geodesic submanifolds of the ambient space. This fact imposes strong restrictions on complete submanifolds of space forms with relative nullity, since the leaves of the minimum relative nullity foliation of a complete submanifold are also complete. The attempt to understand the obstructions imposed by the presence of relative nullity at any point has been of great interest in submanifold theory over the last four decades. For hypersurfaces in space forms with constant index of rela
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