On the energy of a unit vector field

C M Wood

On the energy of a unit vector field

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

The energy of a unit vector field on a Riemannian manifold M is defined to be the energy of the mapping M --> T(1)M, where the unit tangent bundle T(1)M is equipped with the restriction of the Sasaki metric. The constrained variational problem is studied, where variations are confined to unit vector fields, and the first and second variational formulas are derived. The Hopf vector fields on odd-dimensional spheres are shown to be critical points, which are unstable for M = S-5, S-7,..., and an estimate on the index is obtained.

Original language	English
Pages (from-to)	319-330
Number of pages	12
Journal	Geometriae Dedicata
Volume	64
Issue number	3
Publication status	Published - Mar 1997

Keywords

Riemann manifolds
unit vector field
Sasaki metric
Hopf vector fields

Cite this

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title = "On the energy of a unit vector field",

abstract = "The energy of a unit vector field on a Riemannian manifold M is defined to be the energy of the mapping M --> T(1)M, where the unit tangent bundle T(1)M is equipped with the restriction of the Sasaki metric. The constrained variational problem is studied, where variations are confined to unit vector fields, and the first and second variational formulas are derived. The Hopf vector fields on odd-dimensional spheres are shown to be critical points, which are unstable for M = S-5, S-7,..., and an estimate on the index is obtained.",

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year = "1997",

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AU - Wood, C M

PY - 1997/3

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N2 - The energy of a unit vector field on a Riemannian manifold M is defined to be the energy of the mapping M --> T(1)M, where the unit tangent bundle T(1)M is equipped with the restriction of the Sasaki metric. The constrained variational problem is studied, where variations are confined to unit vector fields, and the first and second variational formulas are derived. The Hopf vector fields on odd-dimensional spheres are shown to be critical points, which are unstable for M = S-5, S-7,..., and an estimate on the index is obtained.

AB - The energy of a unit vector field on a Riemannian manifold M is defined to be the energy of the mapping M --> T(1)M, where the unit tangent bundle T(1)M is equipped with the restriction of the Sasaki metric. The constrained variational problem is studied, where variations are confined to unit vector fields, and the first and second variational formulas are derived. The Hopf vector fields on odd-dimensional spheres are shown to be critical points, which are unstable for M = S-5, S-7,..., and an estimate on the index is obtained.

KW - Riemann manifolds

KW - unit vector field

KW - Sasaki metric

KW - Hopf vector fields

M3 - Article

SN - 1572-9168

VL - 64

SP - 319

EP - 330

JO - Geometriae Dedicata

JF - Geometriae Dedicata

IS - 3

ER -