Reconstructions of fronts and caustics depending on a parameter and versality of mappings

A survey of results is presented in the problem of describing up to diffeomorphism rearrangements of fronts and caustics in one-parameter families of general position in spaces of low dimension.

[1]  V. V. Lyčagin.  On Sufficient Orbits of a Group of Contact Diffeomorphisms , 1977 .

[2]  A. Givental,et al.  Mapping of periods and intersection form , 1982 .

[3]  Gordon Wassermann,et al.  Stability of unfoldings in space and time , 1975 .

[4]  Leslie C. Wilson.  Infinitely determined map germs , 1981 .

[5]  Harold Levine,et al.  Singularities of differentiable mappings , 1971 .

[6]  J. Guckenheimer.  Catastrophes and partial differential equations , 1973 .

[7]  J. Dufour,et al.  La théorie des catastrophes. V. Transformées de Legendre et thermodynamique , 1978 .

[8]  Jean Martinet,et al.  Deploiements versels des applications differentiables et classification des applications stables , 1976 .

[9]  B. Plancherel,et al.  La Théorie des catastrophes , 1985 .

[10]  Gordon S Wassermann,et al.  Stability of Unfoldings , 1974 .

[11]  J. Mather.  Stability of C∞ mappings: V, transversality , 1970 .

[12]  A. Weinstein.  Symplectic manifolds and their lagrangian submanifolds , 1971 .

[13]  V. Arnold.  Classification of bimodal critical points of functions , 1974 .

[14]  Alan Weinstein,et al.  Lectures on Symplectic Manifolds , 1977 .

[15]  M. Buchner.  Stability of the cut locus in dimensions less than or equal to 6 , 1977 .

[16]  Characteristic classes of Lagrangian and Legendre manifolds dual to singularities of caustics and wave fronts , 1981 .

[17]  J. Lees.  On the classification of Lagrange immersions , 1976 .

[18]  L. Hörmander.  Fourier integral operators. I , 1995 .

[19]  Vladimir I. Arnold,et al.  INDICES OF SINGULAR POINTS OF 1-FORMS ON A MANIFOLD WITH BOUNDARY, CONVOLUTION OF INVARIANTS OF REFLECTION GROUPS, AND SINGULAR PROJECTIONS OF SMOOTH SURFACES , 1979 .

[20]  Stability of caustics , 1975 .

[21]  V. D. Sedykh.  Singularities of the convex hull of a curve in R3 , 1977 .

[22]  G. Belitskii,et al.  EQUIVALENCE AND NORMAL FORMS OF GERMS OF SMOOTH MAPPINGS , 1978 .

[23]  V. I. Arnol’d.  Lagrange and Legendre cobordisms. II , 1980 .

[24]  V. I. Arnol'd,et al.  Characteristic class entering in quantization conditions , 1967 .

[25]  Martin Golubitsky,et al.  A Theory for Imperfect Bifurcation via Singularity Theory. , 1979 .

[26]  V. I. Arnol'd,et al.  Normal forms for functions near degenerate critical points, the Weyl groups of Ak, Dk, Ek and Lagrangian singularities , 1972 .

[27]  K. Jänich.  Caustics and catastrophes , 1974 .

[28]  Vladimir I. Arnold,et al.  Wave front evolution and equivariant Morse lemma , 1976 .

[29]  General position of equivariant maps , 1977 .

[30]  V. Poénaru,et al.  Versal unfoldings of $G$-invariant functions , 1976 .

[31]  V. M. Zakalyukin,et al.  Singularities of convex hulls of smooth manifolds , 1977 .

[32]  Alan Weinstein,et al.  Lagrangian Submanifolds and Hamiltonian Systems , 1973 .

[33]  John W. Bruce.  A stratification of the space of cubic surfaces , 1980, Mathematical Proceedings of the Cambridge Philosophical Society.

[34]  Jean-Paul Dufour,et al.  Sur la stabilité des diagrammes d'applications différentiables , 1977 .

[35]  D. Trotman.  A Transversality Property Weaker than Whitney (A)‐Regularity , 1976 .

[36]  V. Arnold.  CRITICAL POINTS OF FUNCTIONS ON A MANIFOLD WITH BOUNDARY, THE SIMPLE LIE GROUPS Bk, Ck, AND F4 AND SINGULARITIES OF EVOLUTES , 1978 .

[37]  J. Lees.  Defining Lagrangian immersions by phase functions , 1979 .

[38]  J. Guckenheimer.  Caustics and non-degenerate Hamiltonians☆ , 1974 .

[39]  G. Wassermann.  (r, s)-stable unfoldings and catastrophe theory , 1976 .

[40]  M. Golubitsky.  Contact equivalence for Lagrangian submanifolds , 1975 .

[41]  Louis Solomon.  Invariants of Finite Reflection Groups , 1963, Nagoya mathematical journal.

[42]  Johannes J. Duistermaat,et al.  Oscillatory integrals, lagrange immersions and unfolding of singularities , 1974 .