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Second variation of volume and energy of vector fields. Stability of Hopf vector fields. (English) Zbl 0989.53020

The paper under review deals with second variation of volume and energy of vector fields, and its applications. Given a closed Riemannian manifold \((M, g)\), its tangent sphere bundle \(T^1M\) can be endowed with a natural Riemannian metric \(g^s\) which is called the Sasaki metric. For a unit tangent vector field \(V\) on \(M\), the volume of \(V\), \(F(V)\), is defined as the volume of \(V(M)\) in \((T^1M, g^s)\) and the energy of \(V\) is defined by the energy of the map \(V : (M,g) \to (T^1M, g^s)\).
More generally, denote by \(\chi^1(M)\) and \(\mathcal M\), respectively, the space of unit vector fields and the space of all Riemannian metrics on \(M\). For \(\widetilde g \in \mathcal M\), the energy of the map \(V : (M,\widetilde g) \to (T^1M, g^s)\) is defined by \[ {\overline E}(V,\widetilde g) = \frac{1}{2} \int_M tr(\widetilde g^{-1} V^* g^s) dv {\widetilde g}. \] Denoting by \(E_{\widetilde g}\) the restriction of \(\overline E\) to the slice \(\chi^1(M) \times \{\widetilde g\}\), one obtains a family of functionals \(E_{\widetilde g} : \chi^1(M) \to {\mathbb R}\). The authors prove the second variational formula (Hessian) for the energy functionals \(E_{\widetilde g}\) and volume functional \(F\). As applications, they prove that a Hopf vector field \(V\) on \(S^3\) is stable both for \(F\) and for \(E_{V^*g^s}\) and the nullity of \(V\) as a minimal vector field is \(2\) and as a critical point of \(E_{V^*g^s}\) is \(0\). A Hopf vector field on the sphere \(S^{2m+1}\) is of the form \(JN\), where \(N\) is the unit normal to the sphere and \(J\) is a complex structure on \({\mathbb R}^{2m+2}\). In the higher dimensional case \(m \geq 2\), the authors show that Hopf vector fields on \(S^{2m+1}\) are unstable in contrast to dimension \(3\).

MSC:

53C20 Global Riemannian geometry, including pinching
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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