A Bonnet theorem for isometric immersions into products of space forms

Lira, Jorge H. de; Tojeiro, Ruy; Vitório, Feliciano

doi:10.1007/s00013-010-0183-4

Cited by 30 publications

(39 citation statements)

References 5 publications

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“…We give necessary and sufficient conditions for the existence and uniqueness (modulo isometries of the ambient space) of a isometric immersion of Riemannian manifold into the Riemannian product of two space forms. We note that similar results were independently proved by Lira et al in [4]. Moreover, it was recently shown by Piccione and Tausk in [5] that existence and uniqueness of immersion theorems can be extended to Riemannian manifolds that are "sufficiently homogeneous".…”

Section: Introductionsupporting

confidence: 63%

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Isometric immersions into products of space forms

Kowalczyk

2010

Geom Dedicata

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Section: Introductionsupporting

confidence: 63%

“…where a = c 1 +c 2 4 and b = c 1 −c 2 4 and ∧ associates to two tangent vectors v, w ∈ T p M n+m the endomorphism defined by…”

Section: Preliminariesmentioning

confidence: 99%

Isometric immersions into products of space forms

Kowalczyk

2010

Geom Dedicata

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“…Although this will not be used in the sequel, it is worth mentioning that equations (4) -(8) completely determine an isometric immersion f : M m → Q n ǫ × R up to isometries of Q n ǫ × R (see Corollary 3 of [12]). We now relate the second fundamental forms and normal connections of f andf .…”

Section: Preliminariesmentioning

confidence: 99%

Umbilical Submanifolds of Sⁿ × R

Mendonça¹,

Tojeiro²

2014

Can. j. math.

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Abstract. We give a complete classification of umbilical submanifolds of arbitrary dimension and codimension of S n × R, extending the classification of umbilical surfaces in S 2 × R by Souam and Toubiana as well as the local description of umbilical hypersurfaces in S n ×R by Van der Veken and Vrancken. We prove that, besides small spheres in a slice, up to isometries of the ambient space they come in a two-parameter family of rotational submanifolds whose substantial codimension is either one or two and whose profile is a curve in a totally geodesic S 1 ×R or S 2 ×R, respectively, the former case arising in a one-parameter family. All of them are diffeomorphic to a sphere, except for a single element that is diffeomorphic to Euclidean space. We obtain explicit parametrizations of all such submanifolds. We also study more general classes of submanifolds of S n × R and H n × R. In particular, we give a complete description of all submanifolds in those product spaces for which the tangent component of a unit vector field spanning the factor R is an eigenvector of all shape operators. We show that surfaces with parallel mean curvature vector in S n × R and H n × R having this property are rotational surfaces, and use this fact to improve some recent results by Alencar, do Carmo, and Tribuzy. We also obtain a Dajczer-type reduction of codimension theorem for submanifolds of S n × R and H n × R.

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“…In the statement, U and V stand for ker T and ker(I − T ), respectively. Notice that the third equation in (2) …”

Section: Reduction Of Codimensionmentioning

confidence: 99%

“…We first consider the case (n 1 , n 2 ) = (3,2). This is the case in which T H = 0, and hence k 2 = 0 by (41).…”

Section: Isometric Immersions Such That J Is Totally Geodesic Andf (Mmentioning

confidence: 99%