The exponential map at a cuspidal singularity
-
Vincent Grandjean
and
Daniel Grieser
Abstract
We study spaces with a cuspidal (or horn-like) singularity embedded in a smooth Riemannian manifold and analyze the geodesics in these spaces which start at the singularity. This provides a basis for understanding the intrinsic geometry of such spaces near the singularity. We show that these geodesics combine to naturally define an exponential map based at the singularity, but that the behavior of this map can deviate strongly from the behavior of the exponential map based at a smooth point or at a conical singularity: While it is always surjective near the singularity, it may be discontinuous and non-injective on any neighborhood of the singularity. The precise behavior of the exponential map is determined by a function on the link of the singularity which is an invariant of the induced metric. Our methods are based on the Hamiltonian system of geodesic differential equations and on techniques of singular analysis. The results are proved in the more general natural setting of manifolds with boundary carrying a so-called cuspidal metric.
Funding statement: The first author is very grateful to the Fields Institute for working conditions and support while completing this work. Both authors were supported in part by the Deutsche Forschungsgemeinschaft in the priority program ‘Global Differential Geometry’.
Acknowledgements
The first author is very pleased to thank Carl von Ossietzky Universität Oldenburg for support while visiting twice the second author. Both authors are very grateful to Vincent Naudot for his help and advice on linearization.
References
[1] A. Bernig and A. Lytchak, Tangent spaces and Gromov–Hausdorff limits of subanalytic spaces, J. reine angew. Math. 608 (2007), 1–15. 10.1515/CRELLE.2007.050Search in Google Scholar
[2] L. Birbrair, Local bi-Lipschitz classification of 2-dimensional semialgebraic sets, Houston J. Math. 25 (1999), 453–472. Search in Google Scholar
[3] M. Ghimenti, Geodesics in conical manifolds, Topol. Methods Nonlinear Anal. 25 (2005), 235–261. 10.12775/TMNA.2005.012Search in Google Scholar
[4] D. Grieser, Basics of the b-calculus, Approaches to singular analysis (Berlin 1999), Oper. Theory Adv. Appl. 125, Birkhäuser, Basel (2001), 30–84. 10.1007/978-3-0348-8253-8_2Search in Google Scholar
[5] D. Grieser, Local geometry of singular real analytic surfaces, Trans. Amer. Math. Soc. 355 (2003), no. 4, 1559–1577. 10.1090/S0002-9947-02-03168-9Search in Google Scholar
[6] D. Grieser, Natural differential operator on conic spaces, Discrete Contin. Dyn. Syst. Suppl. (2011), 568–577. Search in Google Scholar
[7] P. Hartman, On local homeomorphisms of Euclidean spaces, Bol. Soc. Mat. Mexi. (2) 5 (1960), 220–241. Search in Google Scholar
[8] A. Hassell, R. B. Melrose and A. Vasy, Spectral and scattering theory for symbolic potentials of order zero, Adv. Math. 181 (2004), 1–87. 10.1016/S0001-8708(03)00020-3Search in Google Scholar
[9] M. W. Hirsch, C. Pugh and M. Shub, Invariant manifolds, Bull. Amer. Math. Soc. 76 (1970), 1015–1019. 10.1090/S0002-9904-1970-12537-XSearch in Google Scholar
[10] R. Melrose, The Atiyah–Patodi–Singer index theorem, A. K. Peters, Newton 1991. Search in Google Scholar
[11] R. Melrose and J. Wunsch, Propagation of singularities for the wave equation on conic manifolds, Invent. Math. 156 (2004), no. 2, 235–299. 10.1007/s00222-003-0339-ySearch in Google Scholar
[12] T. Mostowski, Lipschitz equisingularity, Dissertationes Math. (Rozprawy Mat.) 243 (1985). Search in Google Scholar
[13] A. Parusiński, Lipschitz stratification of subanalytic sets, Ann. Sci. Éc. Norm. Supér. (4) 27 (1994), 661–696. 10.24033/asens.1703Search in Google Scholar
[14] L. Perko, Differential equations and dynamical systems, 3rd ed., Texts Appl. Math. 7, Springer, New York 2001. 10.1007/978-1-4613-0003-8Search in Google Scholar
[15] V. S. Samovol, Linearization of systems of differential equations in the neighborhood of invariant toroidal manifolds, Proc. Moscow Math. Soc. 38 (1979), 187–219. Search in Google Scholar
[16] G. Valette, Lipschitz triangulations, Illinois J. Math. 49 (2005), no. 3, 953–979. 10.1215/ijm/1258138230Search in Google Scholar
[17] G. Valette, On metric types that are definable in an o-minimal structure, J. Symbolic Logic 73 (2008), no. 2, 439–447. 10.2178/jsl/1208359053Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Heat kernel estimates for an operator with a singular drift and isoperimetric inequalities
- The exponential map at a cuspidal singularity
-
Independence of
- Pro unitality and pro excision in algebraic K-theory and cyclic homology
- Kinematic formulas for tensor valuations
- Linear independence of dilogarithmic values
- Positivity in varieties of maximal Albanese dimension
- Effective bounds of linear series on algebraic varieties and arithmetic varieties
- Spectral asymptotics for canonical systems
Articles in the same Issue
- Frontmatter
- Heat kernel estimates for an operator with a singular drift and isoperimetric inequalities
- The exponential map at a cuspidal singularity
-
Independence of
- Pro unitality and pro excision in algebraic K-theory and cyclic homology
- Kinematic formulas for tensor valuations
- Linear independence of dilogarithmic values
- Positivity in varieties of maximal Albanese dimension
- Effective bounds of linear series on algebraic varieties and arithmetic varieties
- Spectral asymptotics for canonical systems
