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Total bending of vector fields on Riemannian manifolds. (English) Zbl 0834.53034
The functional of total bending of globally defined unit vector fields on a Riemannian manifold is devised to measure to what extent they fail to be parallel. The total bending of such a vector field \(X\) is essentially defined as the integral of the squared norm of the covariant differential of \(X\). Many manifolds do not allow parallel vector fields. Aiming at finding vector fields of minimal total bending, some general variational results are derived and the situation on two-dimensional tori is investigated. In this setting rather complete results are obtained.
Reviewer: G.Wiegmink (Köln)

MSC:
53C20 Global Riemannian geometry, including pinching
53A30 Conformal differential geometry (MSC2010)
58E30 Variational principles in infinite-dimensional spaces
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References:
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