Complete submanifolds with relative nullity in space forms

Author(s)

Canevari, Samuel ^[1] ; de Freitas, Guilherme Machado ^[2] ; Guimaraes, Felippe ^[3] ; Manfio, Fernando ^[4] ; dos Santos, Joao Paulo ^[5]

Affiliation

^[1] Univ Fed Sergipe, Campus Prof Alberto Carvalho, BR-49500000 Itabaiana, SE - Brazil

^[2] Inst Mil Engn, Praca Gen Tiburcio, 80 Urca, BR-22290270 Rio De Janeiro, RJ - Brazil

^[3] Univ Sao Paulo, Inst Matemat & Estat, Rua Matao 1010, BR-05508090 Sao Paulo, SP - Brazil

^[4] Univ Sao Paulo, Inst Ciencias Matemat & Comp, Ave Trab Sao Carlense 400, BR-13566590 Sao Carlos, SP - Brazil

^[5] Univ Brasilia, Campus Univ Darcy Ribeiro, BR-70910900 Brasilia, DF - Brazil

• Document type: Journal article
• Source: ANNALS OF GLOBAL ANALYSIS AND GEOMETRY; v. 59, n. 1, p. 81-92, FEB 2021.
• Web of Science Citations: 0

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. (AU)

• FAPESP's process: 19/19494-0 - Virtual immersions, isometric immersions of product manifolds and conformal genuine rigidity
• Grantee: Felippe Soares Guimarães
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• FAPESP's process: 16/23746-6 - Algebraic, topological and analytical techniques in differential geometry and geometric analysis
• Grantee: Paolo Piccione
• Support Opportunities: Research Projects - Thematic Grants