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For $$n\ge 1$$ n ≥ 1 , we exhibit a lower bound for the volume of a unit vector field on $${\mathbb {S}}^{2n+1}\backslash \{\pm p\}$$ S 2 n + 1 \ { ± p } depending on the absolute values of its Poincaré indices around $$\pm p$$ ± p . We determine which vector fields achieve this volume, and discuss the idea of having multiple isolated singularities of arbitrary configurations.

Poincaré index and the volume functional of unit vector fields on punctured spheres