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Publié par | profil-urra-2012 |
Nombre de lectures | 16 |
Langue | English |
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Stability of the characteristic vector field of a
Sasakian manifold
Vincent Borrelli
Dedicated to Bang-Yen Chen for its sixtieth birthday
Abstract. –TheVolumeof a unit vector field is the volume of its image in the
unit tangent bundle.On the standard odd-dimensional spheres, the Hopf vector
fields – that is, unit vector fields tangent to the fiber of any Hopf fibration – are
critical for the volume functional, but they are not alwaysstable. Infact, stability
depends on the radiusrfor every odd dimensionof the sphere :nthere exists a
“critical radius” such that, ifris lower than this radius the Hopf fields are stable on
n
S(r) and conversely.In this article, we show that this phenomenon occurs for the
characteristic vector field of any Sasakian manifold.We then derive two invariants
of a Sasakian manifold, itsE-stability and its stability number.
2000 Mathematics Subject Classification :53C20
Keywords and phrases :Volume, Vector field, Stability, Sasakian manifold.
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General Introduction
Let (M, g) be a oriented Riemanniann manifold, its tangent bundleT M
S
can be endowed with a natural Riemannian metricg ,known as the Sasaki
metric.This metric is defined by :
S
˜ ˜˜ ˜˜ ˜˜ ˜
∀X, Y∈T T M:g(X, Y) =g(dπ(X), dπ(Y)) +g(K(X), K(Y))
whereπ:T M−→Mis the projection andK:T T M−→T Mis the
connector of the Levi-Civita connection∇ofg.LetV:M−→T Mbe a
vector field, thevolumeofVis the volume of the image submanifoldV(M)
S
in (T M, g) :
V ol(V) :=V ol(V(M)).
It can be expressed by the formula :
Z
q
T
V ol(V) =det(Id+∇V◦ ∇V)dvol.
M
In particular,V ol(V)≥V ol(M) with equality if and only if∇V= 0,or, in
other words, ifVis parallel.
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