Stability of the characteristic vector field of a Sasakian manifold

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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. – The Volume of 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 always stable. In fact, stability depends on the radius r of the sphere : for every odd dimension n there exists a “critical radius” such that, if r is lower than this radius the Hopf fields are stable on Sn(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, its E-stability and its stability number. 2000 Mathematics Subject Classification : 53C20 Keywords and phrases : Volume, Vector field, Stability, Sasakian manifold. 1 General Introduction Let (M, g) be a oriented Riemanniann manifold, its tangent bundle TM can be endowed with a natural Riemannian metric gS , known as the Sasaki metric.This metric is defined by : ?X˜, Y˜ ? TTM : gS(X˜, Y˜ ) = g(dpi(X˜), dpi(Y˜ )) + g(K(X˜),K(Y˜ )) where pi : TM ?? M is the projection and K : TTM ?? TM is the connector of the

  • standard sasakian

  • vector field

  • ?? ?

  • minimum volume

  • hopf vector

  • fields

  • odd-dimensional spheres

  • ?k ?t?a ?


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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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