Geometry of submanifolds with respect to ambient vector fields
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Geometry of submanifolds with respect to ambient vector fields Fernando Manfio1 · Ruy Tojeiro1 · Joeri Van der Veken2 Received: 27 September 2019 / Accepted: 15 February 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Given a Riemannian manifold N n and Z ∈ 𝔛(N) , an isometric immersion f ∶ M m → N n is said to have the constant ratio property with respect to Z either if the tangent component ZTf of Z vanishes identically or if ZTf vanishes nowhere and the ratio ‖Z⟂f ‖∕‖ZTf ‖ between the lengths of the normal and tangent components of Z is constant along M m . It has the principal direction property with respect to Z if ZTf is an eigenvector of all shape operators of f at all points of M m . In this article, we study isometric immersions f ∶ M m → N n of arbitrary codimension that have either the constant ratio or the principal direction property with respect to distinguished vector fields Z on space forms, product spaces 𝕊n × ℝ and ℍn × ℝ , where 𝕊n and ℍn are the n-dimensional sphere and hyperbolic space, respectively, and, more generally, on warped products I ×𝜌 ℚn𝜖 of an open interval I ⊂ ℝ and a space form ℚn𝜖 . Starting from the observation that these properties are invariant under conformal changes of the ambient metric, we provide new characterizations and classification results of isometric immersions that satisfy either of those properties, or both of them simultaneously, for several relevant instances of Z as well as simpler descriptions and proofs of some known ones for particular cases of Z previously considered by many authors. Our methods also allow us to classify Euclidean submanifolds with the property that the normal components of their position vector fields are parallel with respect to the normal connection, and to give alternative descriptions to those in Chen (J Geom 74(1–2): 61–77, 2002) of Euclidean submanifolds whose tangent or normal components of their position vector fields have constant length. Keywords Constant ratio property · Principal direction property · Radial vector field · Killing vector field · Conformal Killing vector field · Loxodromic isometric immersion Mathematics Subject Classification 53 B25 Parts of this work were carried out while J. Van der Veken visited Universidade de São Paulo in the framework of the Young Researchers Summer Program 2018, and while Ruy Tojeiro visited the KULeuven partially supported by Aucani (USP) Public Notice 967/2018. J. Van der Veken is supported by project 3E160361 of the KU Leuven Research Fund and by EOS project G0H4518N of the Belgian government. Ruy Tojeiro is supported by Fapesp Grant 2016/23746-6 and CNPq Grant 303002/2017-4. * Fernando Manfio [email protected] Extended author information available on the last page of the article
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1 Introduction Let M m and N n be Riemannian manifolds of dimensions m and n, respectively. Given a vector field Z on N n , it is a natural problem to investigate the isometric i
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