Search results for "53C25"
showing 2 items of 2 documents
Manifolds with vectorial torsion
2015
Abstract The present note deals with the properties of metric connections ∇ with vectorial torsion V on semi-Riemannian manifolds ( M n , g ) . We show that the ∇-curvature is symmetric if and only if V ♭ is closed, and that V ⊥ then defines an ( n − 1 ) -dimensional integrable distribution on M n . If the vector field V is exact, we show that the V-curvature coincides up to global rescaling with the Riemannian curvature of a conformally equivalent metric. We prove that it is possible to construct connections with vectorial torsion on warped products of arbitrary dimension matching a given Riemannian or Lorentzian curvature—for example, a V-Ricci-flat connection with vectorial torsion in di…
Minimal unit vector fields
2002
We compute the first variation of the functional that assigns each unit vector field the volume of its image in the unit tangent bundle. It is shown that critical points are exactly those vector fields that determine a minimal immersion. We also find a necessary and sufficient condition that a vector field, defined in an open manifold, must fulfill to be minimal, and obtain a simpler equivalent condition when the vector field is Killing. The condition is fulfilled, in particular, by the characteristic vector field of a Sasakian manifold and by Hopf vector fields on spheres.