Abstract
We discuss new sufficient conditions under which an affine manifold \((M,\nabla )\) is geodesically connected. These conditions are shown to be essentially weaker than those discussed in groundbreaking work by Beem and Parker and in recent work by Alexander and Karr, with the added advantage that they yield an elementary proof of the main result.
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Notes
It is well known, again due to the absence of a Hopf–Rinow-like result, that compactness and geodesic completeness are unrelated in Lorentzian manifolds (cf. Example 7.17 in [20]).
We remind the reader that a connected complete Riemannian manifold is said to be a Wiedersehen manifold if \(\mathrm{Conj}(p)\) is just one point for every \(p \in M\). The importance of these manifolds arises due to the so-called (original) Blaschke conjecture, which posited that any Wiedersehen manifolds of \(dim\ M=2\) are isometric to a round sphere \(\mathbb {S}^2\). This has been proven by Green [15], and the analogue statement in higher dimensions has been proven by Berger, Kazdan, Weinstein and Yang [10, 18, 23, 24].
Here and hereafter, smooth means \(C^{\infty }\) and manifolds for us always mean real, smooth, finite-dimensional, Hausdorff, second-countable manifolds.
As proven in [16], this condition is equivalent to precompactness of the holonomy group of (M, g).
In [22], the authors actually require that (M, g) be Ricci-flat, but their integral formula is easily seen to apply when one only has \(\mathrm{Ric}(V,V)\le 0\).
When \(F=\exp _p\), we chose \({\mathcal {V}}\) to be a normal neighborhood of p.
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Acknowledgements
The authors are partially supported by the Project MTM2016-78807-C2-2-P (Spanish MINECO with FEDER funds).
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Costa e Silva, I.P., Flores, J.L. Geodesic connectedness of affine manifolds. Annali di Matematica 200, 1135–1148 (2021). https://doi.org/10.1007/s10231-020-01028-8
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DOI: https://doi.org/10.1007/s10231-020-01028-8