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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 \(3\).

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 \(3\).

Reviewer: Gabjin Yun (Kyung-Ki, Yong-in)

### MSC:

53C20 | Global Riemannian geometry, including pinching |

53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |