Unit vector fields on antipodally punctured spheres: big index, big volume. (English) Zbl 1158.53023

The authors establish a lower bound for the volume of a unit vector field \(\overrightarrow{v}\) defined on \(S^{n}\backslash\{\pm\, x\},\) \(n=2,3.\) This lower bound depends on the sum of the absolute values of the indices of \(\overrightarrow{v}\) at \(x\) and \(-x.\) This result is a preliminary one for future studies concerning relations between the volume of unit vector fields and their indexes around isolated singularities.


53C20 Global Riemannian geometry, including pinching
57R25 Vector fields, frame fields in differential topology
53C12 Foliations (differential geometric aspects)
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