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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.

MSC:

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

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