Abstract
It is known that Blaschke manifolds (where injectivity radius equals diameter) are Besse manifolds (where all geodesics are closed). We show that Besse manifolds with sufficiently many diameter realizing directions are Blaschke. We also provide bounds in terms of diameter on the length of the shortest closed geodesic for pinched curvature metrics on simply connected manifolds.
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Acknowledgements
The authors would like to thank Wolfgang Ziller for suggesting the reference [4] and helpful discussions about short closed geodeiscs in the pinched curvature setting. Vargas Pallete was supported by DMS-2001997.
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Adelstein, I., Pallete, F.V. Besse Projective Spaces with Many Diameters. J Geom Anal 33, 277 (2023). https://doi.org/10.1007/s12220-023-01337-3
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DOI: https://doi.org/10.1007/s12220-023-01337-3