Convex neighborhoods for Lipschitz connections and sprays

详细信息    查看全文

• 作者：E. Minguzzi
• 关键词：Lipschitz connections ; Exponential map ; Convex neighborhood ; Distance function ; Low differentiability ; 53B15 ; 26A16 ; 53B40 ; 83Cxx
• 刊名：Monatshefte f篓鹿r Mathematik
• 出版年：2015
• 出版时间：August 2015
• 年：2015
• 卷：177
• 期：4
• 页码：569-625
• 全文大小：849 KB
• 参考文献：1.Akbar-Zadeh, H.: Sur les espaces de Finsler a courbures sectionnelles constantes. Acad. Roy. Belg. Bull. Cl. Sci. 74, 281-22 (1988)MathSciNet
  2.Ambrose, W., Palais, R.S., Singer, I.M.: Sprays. An. Acad. Brasil. Ciênc. 32(2), 163-78 (1960)MathSciNet
  3.Andersson, L., Galloway, G.J., Howard, R.: The cosmological time function. Class. Quantum Grav. 15, 309-22 (1998)MathSciNet View Article
  4.Antonelli, P.L., Ingarden, R.S., Matsumoto, M.: The theory of sprays and Finsler spaces with applications in physics and biology. Springer, Dordrecht (1993)View Article
  5.Bao, D., Chern, S.S., Shen, Z.: An introduction to Riemann-Finsler geometry. Springer, New York (2000)View Article
  6.Beem, J.K.: Indefinite Finsler spaces and timelike spaces. Can. J. Math. 22, 1035-039 (1970)MathSciNet View Article
  7.Beem, J.K.: Characterizing Finsler spaces which are pseudo-Riemannian of constant curvature. Pacific J. Math. 64, 67-7 (1976)MathSciNet View Article
  8.Beem, J.K., Ehrlich, P.E., Easley, K.L.: Global Lorentzian geometry. Marcel Dekker Inc., New York (1996)
  9.Berestovskij, V.N., Nikolaev, I.G., Reshetnyak, Y.G.: Geometry IV: non-regular Riemannian geometry. Springer, Berlin (1993)
  10.Buttazzo, G., Giaquinta, M.: One-dimensional variational problems. Oxford University Press, Oxford (1998)
  11.Candela, A.M., Flores, J.L., Sánchez, M.: Global hyperbolicity and Palais-Smale condition for action functionals in stationary spacetimes. Adv. Math. 218, 515-36 (2008)MathSciNet View Article
  12.do Carmo, M.: Riemannian geometry. Birkh?user, Boston (1992)View Article
  13.Cartan, H.: Differential calculus. Hermann, Paris (1971)
  14.Chen, B.L., LeFloch, P.G.: Injectivity radius of Lorentzian manifolds. Commun. Math. Phys. 278, 679-13 (2008)MathSciNet View Article
  15.Chru?ciel, P.T.: Elements of causality theory (2011). ArXiv:-110.-706v1
  16.Chru?ciel, P.T., Grant, J.D.E.: On Lorentzian causality with continuous metrics. Class. Quantum Grav. 29, 145001 (2012)View Article
  17.Clarke, F.H.: On the inverse function theorem. Pacific J. Math. 64, 97-02 (1976)MathSciNet View Article
  18.DeTurck, D.M., Kazdan, J.L.: Some regularity theorems in Riemannian geometry. Ann. Sci. école Norm. Sup. 4(14), 249-60 (1981)MathSciNet
  19.Dieudonné, J.: Treatise on Analysis. Foundations of modern analysis. Academic Press, New York (1969)
  20.Dolecki, S., Greco, G.H.: Amazing oblivion of Peano’s legacy (2012)
  21.Esser, M., Shisha, O.: A modified differentiation. Am. Math. Mon. 71, 904-06 (1964)MathSciNet View Article
  22.Evans, L.C.: Partial differential equations. American Mathematical Society, Providence (1998)
  23.Fan, L., Liu, S., Gao, S.: Generalized monotonicity and convexity of non-differentiable functions. J. Math. Anal. Appl. 279, 276-89 (2003)
  24.Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93, 418-91 (1959)MathSciNet View Article
  25.Foertsch, T.: Ball versus distance convexity of metric spaces. Beitr?ge Algebra Geom. 45, 481-00 (2004)MathSciNet
  26.Gallot, S., Hulin, D., Lafontaine, J.: Riemannian geometry. Springer, Berlin (1987)View Article
  27.Garg, K.M.: Theory of differentiation. Wiley, New York (1998)
  28.Harris, S.G.: A triangle comparison theorem for Lorentz manifolds. Indiana Univ. Math. J. 31, 289-08 (1982)MathSciNet View Article
  29.Hartman, P.: On the local uniqueness of geodesics. Am. J. Math. 72, 723-30 (1950)View Article
  30.Hartman, P.: On geodesic coordinates. Am. J. Math. 73, 949-54 (1951)View Article
  31.Hartman, P.: Ordinary differential equations. Wiley, New York (1964)
  32.Hartman, P.: Remarks on geodesics. Proc. Am. Math. Soc. 89, 467-72 (1983)View Article
  33.Hartman, P., Wintner, A.: On the problems of geodesics in the small. Am. J. Math. 73, 132-48 (1951)MathSciNet View Article
  34.Hawking, S.W., Ellis, G.F.R.: The large scale structure of space-time. Cambridge University Press, Cambridge (1973)View Article
  35.Krantz, S.G., Parks, H.R.: The implicit function theorem: history, theory, and applications. Birkh?user, Boston (2000)
  36.Kunzinger, M., Steinbauer, R., Stojkovi?, M.: The exponential map of a \(C^{1,1}\) -metric. Differential Geom. Appl. 34, 14-4 (2014). ArXiv:-306.-776v1
  37.Kunzinger, M., Steinbauer, R., Stojkovi?, M., Vickers, J.A.: A regularisation approach to causality theory for \(C^{1,1}\) -Lorentzian metrics. Gen. Relativ. Gravit. 46, 1738 (2014). ArXiv:-310.-404v2
  38.Lang, S.: Differential and Riemannian manifolds. Springer, New York (1995)View Article
  39.Lang, S.: Fundamentals of differential geometry. Springer, New York (1999)View Article
  40.Leach, E.B.: A note on inverse function theorems. Proc. Am. Math. Soc. 12, 694-97 (1961)MathSciNet View Article
  41.Leach, E.B.: On a related function theorem. Proc. Am. Math. Soc. 14, 687-89 (1963)MathSciNet View Article
  42.Matsumoto, M.: Foundations of Finsler geometry and special Finsler spaces. Kaseisha
• 作者单位：E. Minguzzi (1)
  
  1. Dipartimento di Matematica e Informatica “U. Dini- Università degli Studi di Firenze, Via S. Marta 3, 50139, Florence, Italy
  
• 刊物主题：Mathematics, general;
• 出版者：Springer Vienna
• ISSN：1436-5081

文摘

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-Lindel?f 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.