Abstract
We generalize the Fenchel theorem to strong spacelike (which means that the tangent vector and the curvature vector span a spacelike 2-plane at each point) closed curves with index 1 in the 3-dimensional Lorentz space, showing that the total curvatures must be less than or equal to \(2\pi \). A similar generalization of the Fary-Milnor theorem is also obtained. We establish the Crofton formula on the de Sitter 2-sphere which implies the above results.
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References
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Ye, N., Ma, X. & Wang, D. The Fenchel-type inequality in the 3-dimensional Lorentz space and a Crofton formula. Ann Glob Anal Geom 50, 249–259 (2016). https://doi.org/10.1007/s10455-016-9510-8
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DOI: https://doi.org/10.1007/s10455-016-9510-8