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Convex neighborhoods for Lipschitz connections and sprays

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Abstract

We establish that over a \(C^{2,1}\) manifold the exponential map of any Lipschitz connection or spray determines a local Lipeomorphism and that, furthermore, reversible convex normal neighborhoods do exist. To that end we use the method of Picard-Lindelöf approximation to prove the strong differentiability of the exponential map at the origin and hence a version of Gauss’ Lemma which does not require the differentiability of the exponential map. Contrary to naive differential degree counting, the distance functions are shown to gain one degree and hence to be \(C^{1,1}\). As an application to mathematical relativity, it is argued that the mentioned differentiability conditions can be considered the optimal ones to preserve most results of causality theory. This theory is also shown to be generalizable to the Finsler spacetime case. In particular, we prove that the local Lorentzian(-Finsler) length maximization property of causal geodesics in the class of absolutely continuous causal curves holds already for \(C^{1,1}\) spacetime metrics. Finally, we study the local existence of convex functions and show that arbitrarily small globally hyperbolic convex normal neighborhoods do exist.

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Notes

  1. Please notice that according to [32] claim III in [33] is incorrect.

  2. There are well known difficulties in this generalization. They are related to the fact that sectional curvature bounds imply constant curvature [28, 50]. These problems could be sidestepped imposing only bounds on the sectional curvature of timelike planes [28].

  3. One could ask whether every Lipschitz function is strongly differentiable almost everywhere. The answer is negative already for functions defined on the real line [27], Sect. 14.4.1].

  4. The fact that \(N_1\) and \(N_2\) can be chosen to be open sets such that both \(f|_{N_1}\) and \(g\) are Lipschitz follows from Leach’s original formulation plus (i). The statement in the last paragraph is not contained in Leach’s original formulation but can be found in its proof.

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Acknowledgments

After this paper was posted on the preprint archive (1308.6675) another work by M. Kunzinger, R. Steinbauer, M. Stojković and J. A. Vickers was posted which obtained some of the results contained in this work through a regularization scheme for \(C^{1,1}\) metrics [37]. The approaches of this and that work nicely complement each other. However, the results of this work seem more general (e.g. they apply to Finsler geometry) and stronger (e.g. we prove strong differentiability of the exponential map at he origin). Also the proofs we provide are technically more elementary. The reader is referred to [37] for a comparison. I thank Piotr Chruściel for some useful suggestions.

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  1. Dipartimento di Matematica e Informatica “U. Dini”, Università degli Studi di Firenze, Via S. Marta 3, 50139, Florence, Italy

    E. Minguzzi

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Correspondence to E. Minguzzi.

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Communicated by P. Chrusciel.

This work has been partially supported by GNFM of INDAM.

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Minguzzi, E. Convex neighborhoods for Lipschitz connections and sprays. Monatsh Math 177, 569–625 (2015). https://doi.org/10.1007/s00605-014-0699-y

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