Codazzi tensors and reducible submanifolds

Bivens, Irl C.

doi:10.1007/bfb0088865

Cited by 7 publications

(6 citation statements)

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“…The name comes from the fact that the Codazzi equations for a hypersurface in R n simply say that the second fundamental form is a Codazzi tensor. Both topological and geometric consequences of the existence of a nontrivial Codazzi tensor field on a Riemannian manifold have been studied in [3,10].…”

Section: Introductionmentioning

confidence: 99%

Left invariant Riemannian metrics with harmonic curvature are Ricci-parallel in solvable Lie groups and Lie groups of dimension ≤6

Aberaouze

Boucetta

2022

Journal of Geometry and Physics

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Section: Introductionmentioning

confidence: 99%

Left invariant Riemannian metrics with harmonic curvature are Ricci-parallel in solvable Lie groups and Lie groups of dimension ≤6

Aberaouze

Boucetta

2022

Journal of Geometry and Physics

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“…Locally conformally flat multidimensional cosmological models leads to warped and multiply warped product decompositions of the spacetime in many cases, the mathematical structure behind multidimensional models (see for example [2,24] and the references therein for more information on Codazzi tensors).…”

Section: Jcap12(2004)008mentioning

confidence: 99%

“…In what follows we restrict ourselves to multiply warped spaces with one-dimensional base whose fibres are all Riemannian manifolds as in (1), (2). Therefore, we consider a Lorentzian manifold (M, g) with the underlying structure of a multiply warped product space of the form…”

Section: Jcap12(2004)008mentioning

confidence: 99%

Locally conformally flat multidimensional cosmological models and generalized Friedmann–Robertson–Walker spacetimes

Brozos‐Vázquez¹,

Garcı́a-Rı́o²,

Vázquez-Lorenzo³

2004

J. Cosmol. Astropart. Phys.

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“…Suppose M is a compact oriented Riemannian manifold without boundary isometrically immersed into R^ and 5 is a parallel distribution on M. Then each leaf of g is a submanifold of R^ and given p G M we let r¡(p) denote the mean curvature normal vector at/? for the corresponding immersed leaf of g. In [1] we showed that r , ,2…”

Section: Orthogonal Geodesic and Minimal Distributions By Irl Bivensmentioning

confidence: 99%

Orthogonal Geodesic and Minimal Distributions

Bivens¹

1983

Transactions of the American Mathematical Society

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Abstract. Let 8 be a smooth distribution on a Riemannian manifold M with $ the orthogonal distribution. We say that 5 is geodesic provided g is integrable with leaves which are totally geodesic submanifolds of M. The notion of minimality of a submanifold of M may be defined in terms of a criterion involving any orthonormal frame field tangent to the given submanifold. If this criterion is satisfied by any orthonormal frame field tangent to ip then we say ip is minimal. Suppose that 8 and § are orthogonal geodesic and minimal distributions on a submanifold of Euclidean space. Then each leaf of 3 is also a submanifold of Euclidean space with mean curvature normal vector field r¡. We show that the integral of | ij |2 over M is bounded below by an intrinsic constant and give necessary and sufficient conditions for equality to hold.We study the relationships between the geometry of M and the integrability of §. For example, if 3 and § are orthogonal geodesic and minimal distributions on a space of nonnegative sectional curvature then § is integrable iff 3 and § are parallel distributions. Similarly if M" has constant negative sectional curvature and dim § = 2 < n then § is not integrable. If 3 is geodesic and § is integrable then we characterize the local structure of the Riemannian metric in the case that the leaves of § are flat submanifolds of A/, with parallel second fundamental form.

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Codazzi tensors and reducible submanifolds

Cited by 7 publications

References 1 publication

Left invariant Riemannian metrics with harmonic curvature are Ricci-parallel in solvable Lie groups and Lie groups of dimension ≤6

Left invariant Riemannian metrics with harmonic curvature are Ricci-parallel in solvable Lie groups and Lie groups of dimension ≤6

Locally conformally flat multidimensional cosmological models and generalized Friedmann–Robertson–Walker spacetimes

Orthogonal Geodesic and Minimal Distributions

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