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Total bending of flows with mean curvature correction. (English) Zbl 0995.53023
The author introduces the functional $D(\vec{v}) = \int_M \left( \|\nabla \vec{v} \|^2 +\;(n-1) (n-3) \|\vec{H}_{\vec{v}_{\bot}}\|^2 \right) dM,$ on the unit vector fields $$\vec v$$ of a closed Riemannian manifold $$M$$, where $$\vec{H}_{\vec{v}_{\bot}}$$ is the mean curvature vector of the distribution $$\vec{v}_{\bot}$$, orthogonal to $$\vec{v}$$. He shows that Hopf flows uniquely minimize $$D$$ on odd-dimensional unit spheres. On $$S^3$$ the functional $$D$$ coincides, up to a constant, with the total bending of a unit vector field as defined by G. Wiegmink. Therefore it follows that Hopf flows minimize $$B$$ on $$S^3$$ and, moreover, these are the unique absolute minima of $$B$$.
The paper also extends a result of H. Gluck and W. Ziller on the lower bound for the volume of vector fields on $$S^3.$$

##### MSC:
 53C20 Global Riemannian geometry, including pinching
##### Keywords:
Hopf flows; mean curvature of a distribution; total bending
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