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Let  M  be an oriented three-dimensional Riemannian manifold. We define a notion of vorticity of local sections of the bundleSO⁢(M)→M{\mathrm{SO}(M)\rightarrow M}of all its positively oriented orthonormal tangent frames. When  M  is a space form, we relate the concept to a suitable invariant split pseudo-Riemannian metric onIsoo⁢(M)≅SO⁢(M){\mathrm{Iso}_{o}(M)\cong\mathrm{SO}(M)}: A local section has positive vorticity if and only if it determines a space-like submanifold. In the Euclidean case we find explicit homologically volume maximizing sections using a split special Lagrangian calibration. We introduce the concept of optimal frame vorticity and give an optimal screwed global section for the three-sphere. We prove that it is also homologically volume maximizing (now using a common one-point split calibration). Besides, we show that no optimal section can exist in the Euclidean and hyperbolic cases.

A split special Lagrangian calibration associated with frame vorticity