Convex neighborhoods for Lipschitz connections and sprays

Abstract

We establish that over a C^{2,1} manifold the exponential map of any Lipschitz connection or spray determines a local Lipeomophism and that, furthermore, reversible convex normal neighborhoods do exist. To that end we use the method of Picard-Lindelof 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.

56 pages. v2: modified comment on Gauss lemma under C^2 differentiability conditions on Sect. 1.3. Matches published version

References in corpus (7)

• Singularity Theorems and Their Consequences
• The causal hierarchy of spacetimes
• Light cones in Finsler spacetime
• Limit curve theorems in Lorentzian geometry
• The causal ladder and the strength of K-causality. II
• A regularisation approach to causality theory for $C^{1,1}$-Lorentzian metrics
• The exponential map of a $C^{1,1}$-metric

Cited by in corpus (49)

• The 1965 Penrose singularity theorem
• Causality theory for closed cone structures with applications
• On the definition and examples of cones and Finsler spacetimes
• Light cones in Finsler spacetime
• Global hyperbolicity for spacetimes with continuous metrics
• Hawking's singularity theorem for $C^{1,1}$-metrics
• Singularity theorems for $C^1$-Lorentzian metrics
• The Penrose singularity theorem in regularity $C^{1,1}$
• Standard static Finsler spacetimes
• Inextendibility of spacetimes and Lorentzian length spaces
• The Hawking-Penrose singularity theorem for $C^{1,1}$-Lorentzian metrics
• Raychaudhuri equation and singularity theorems in Finsler spacetimes
• Maximizers in Lipschitz spacetimes are either timelike or null
• The future is not always open
• Globally hyperbolic spacetimes can be defined without the 'causal' condition
• Length structures on manifolds with continuous Riemannian metrics
• The singularity theorems of General Relativity and their low regularity extensions
• A Lorentzian analog for Hausdorff dimension and measure
• Every Lipschitz metric has $C^1$-geodesics
• The Hawking-Penrose singularity theorem for $C^1$-Lorentzian metrics
• The global uniqueness and $C^1$-regularity of geodesics in expanding impulsive gravitational waves
• An equivalence of Finslerian relativistic theories
• Affine sphere relativity
• On geodesics in low regularity
• Geometry of weighted Lorentz-Finsler manifolds I: Singularity theorems
• Lorentz meets Lipschitz
• A Note on the Gannon-Lee Theorem
• Affine sphere spacetimes which satisfy the relativity principle
• Cut-and-paste for impulsive gravitational waves with $\Lambda$: The geometric picture
• Volume comparison for $C^{1,1}$ metrics
• Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems
• Huygens' envelope principle in Finsler spaces and analogue gravity
• Gluing constructions for Lorentzian length spaces
• Special coordinate systems in pseudo-Finsler geometry and the equivalence principle
• Geometry of weighted Lorentz-Finsler manifolds II: A splitting theorem
• Synthetic versus distributional lower Ricci curvature bounds
• Cut-and-paste for impulsive gravitational waves with $\Lambda$: The mathematical analysis
• Green Operators in Low Regularity Spacetimes and Quantum Field Theory
• On the regularity of Cauchy hypersurfaces and temporal functions in closed cone structures
• Inequalities from Lorentz-Finsler norms
• Spacetimes categories and disjointness for algebraic quantum field theory
• Wind-Finslerian structure of black holes
• Comparison theorems on weighted Finsler manifolds and spacetimes with $\epsilon$-range
• A Scale-Dependent Distance Functional between Past Light Cones in Cosmology
• The equality of mixed partial derivatives under weak differentiability conditions
• Fixed energy solutions to the Euler-Lagrange equations of an indefinite Lagrangian with affine Noether charge
• The representation of spacetime through time functions
• Concavity of spacetimes
• How Dark is Dark Energy? A Lightcones Comparison Approach