Conformally flat submanifolds with flat normal bundle
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© Springer-Verlag GmbH Germany, part of Springer Nature 2019
M. Dajczer · C.-R. Onti · Th. Vlachos
Conformally flat submanifolds with flat normal bundle In memory of Manfredo do Carmo Received: 19 January 2019 / Accepted: 10 October 2019 Abstract. We prove that any conformally flat submanifold with flat normal bundle in a conformally flat Riemannian manifold is locally holonomic, that is, admits a principal coordinate system. As one of the consequences of this fact, it is shown that the Ribaucour transformation can be used to construct an associated large family of immersions with induced conformally flat metrics holonomic with respect to the same coordinate system.
A main task in conformal geometry is the study of submanifolds of conformally flat Riemannian manifolds with induced conformally flat metrics. A Riemannian manifold M n is said to be conformally flat if each point lies in an open neighborhood conformal to an open subset of Euclidean space Rn . This is always the case for manifolds endowed with metrics of constant sectional curvature. Even if they belong to the realm of conformal geometry, for reason of simplicity most of the results in this paper are stated for submanifolds of Euclidean space. Nevertheless, they hold true when the ambient space is just a conformally flat manifold. E. Cartan [1] proved that a hypersurface f : M n → Rn+1 , n ≥ 4, is conformally flat if and only if at each point there is a principal curvature of multiplicity at least n−1. If M n is free of flat points, then f (M) is locally foliated by (n−1)-dimensional umbilical submanifolds of Rn+1 , or equivalently, we have that f (M) is enveloped by a one-parameter family of umbilical hypersurfaces of the ambient space. Moore [14] extended Cartan’s result to submanifolds of higher codimension. He showed that an isometric immersion of a conformally flat manifold f : M n → Rn+ p of dimension n ≥ 4 and codimension p ≤ n − 3 has a principal normal vector of multiplicity at least n − p ≥ 3 at each point. Recall that a normal vector η ∈ N f M(x) is called a principal normal of f at x ∈ M n with multiplicity s if the tangent subspace defined as E η (x) = X ∈ Tx M : α f (X, Y ) = X, Y η for all Y ∈ Tx M M. Dajczer · C.-R. Onti: IMPA, Estrada Dona Castorina, 110, Rio de Janeiro 22460-320, Brazil. e-mail: [email protected] C.-R. Onti: e-mail: [email protected] Th. Vlachos (B): Department of Mathematics, University of Ioannina, Ioannina, Greece. e-mail: [email protected] Mathematics Subject Classification: 53C40 · 53C42
https://doi.org/10.1007/s00229-019-01158-1
M. Dajczer et al.
in terms of the second fundamental form α f : T M × T M → N f M of the immersion, satisfies dim E η (x) = s > 0. Clearly, principal normals are a natural generalization to submanifolds of higher codimension of principal curvatures of hypersurfaces. A smooth normal vector field η ∈ N f M to an isometric immersion f : M n → N R is called a principal normal vector field with multiplicity s if dim E η (x) = s > 0 is constant, in which case the distribution x ∈ M
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